{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0001","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=0x+(0)，求 x→0 的去心極限。","answer":{"kind":"short_text","value":"0","explanation_zh":"去心極限=0。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082001,"parameters":{"m":0,"b":0,"a":0,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"value_b\",\"0\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0002","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-2x+(-11)，求 x→0 的去心極限。","answer":{"kind":"short_text","value":"-11","explanation_zh":"去心極限=-11。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082002,"parameters":{"m":-2,"b":-11,"a":0,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"value_b\",\"-11\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0003","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=3x+(-10)，求 x→3 的去心極限。","answer":{"kind":"short_text","value":"-1","explanation_zh":"去心極限=-1。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082003,"parameters":{"m":3,"b":-10,"a":3,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"3\",\"-10\",\"3\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0004","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-4x+(-9)，求 x→4 的去心極限。","answer":{"kind":"short_text","value":"-25","explanation_zh":"去心極限=-25。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082004,"parameters":{"m":-4,"b":-9,"a":4,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-4\",\"-9\",\"4\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0005","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=5x+(-8)，求 x→1 的去心極限。","answer":{"kind":"short_text","value":"-3","explanation_zh":"去心極限=-3。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082005,"parameters":{"m":5,"b":-8,"a":1,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"5\",\"-8\",\"1\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0006","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-6x+(-7)，求 x→2 的去心極限。","answer":{"kind":"short_text","value":"-19","explanation_zh":"去心極限=-19。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082006,"parameters":{"m":-6,"b":-7,"a":2,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-6\",\"-7\",\"2\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0007","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=7x+(-6)，求 x→3 的去心極限。","answer":{"kind":"short_text","value":"15","explanation_zh":"去心極限=15。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082007,"parameters":{"m":7,"b":-6,"a":3,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"7\",\"-6\",\"3\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0008","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-8x+(-5)，求 x→4 的去心極限。","answer":{"kind":"short_text","value":"-37","explanation_zh":"去心極限=-37。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082008,"parameters":{"m":-8,"b":-5,"a":4,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-8\",\"-5\",\"4\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0009","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=9x+(-4)，求 x→1 的去心極限。","answer":{"kind":"short_text","value":"5","explanation_zh":"去心極限=5。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082009,"parameters":{"m":9,"b":-4,"a":1,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"9\",\"-4\",\"1\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0010","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-10x+(-3)，求 x→2 的去心極限。","answer":{"kind":"short_text","value":"-23","explanation_zh":"去心極限=-23。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082010,"parameters":{"m":-10,"b":-3,"a":2,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-10\",\"-3\",\"2\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0011","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=11x+(-2)，求 x→3 的去心極限。","answer":{"kind":"short_text","value":"31","explanation_zh":"去心極限=31。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082011,"parameters":{"m":11,"b":-2,"a":3,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"11\",\"-2\",\"3\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0012","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-12x+(-1)，求 x→4 的去心極限。","answer":{"kind":"short_text","value":"-49","explanation_zh":"去心極限=-49。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082012,"parameters":{"m":-12,"b":-1,"a":4,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-12\",\"-1\",\"4\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0013","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=13x+(0)，求 x→1 的去心極限。","answer":{"kind":"short_text","value":"13","explanation_zh":"去心極限=13。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082013,"parameters":{"m":13,"b":0,"a":1,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"13\",\"0\",\"1\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0014","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-14x+(1)，求 x→2 的去心極限。","answer":{"kind":"short_text","value":"-27","explanation_zh":"去心極限=-27。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082014,"parameters":{"m":-14,"b":1,"a":2,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-14\",\"1\",\"2\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0015","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=15x+(2)，求 x→3 的去心極限。","answer":{"kind":"short_text","value":"47","explanation_zh":"去心極限=47。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082015,"parameters":{"m":15,"b":2,"a":3,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"15\",\"2\",\"3\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0016","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-16x+(3)，求 x→4 的去心極限。","answer":{"kind":"short_text","value":"-61","explanation_zh":"去心極限=-61。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082016,"parameters":{"m":-16,"b":3,"a":4,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-16\",\"3\",\"4\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0017","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=17x+(4)，求 x→1 的去心極限。","answer":{"kind":"short_text","value":"21","explanation_zh":"去心極限=21。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082017,"parameters":{"m":17,"b":4,"a":1,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"17\",\"4\",\"1\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0018","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-18x+(5)，求 x→2 的去心極限。","answer":{"kind":"short_text","value":"-31","explanation_zh":"去心極限=-31。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082018,"parameters":{"m":-18,"b":5,"a":2,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-18\",\"5\",\"2\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0019","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=19x+(6)，求 x→3 的去心極限。","answer":{"kind":"short_text","value":"63","explanation_zh":"去心極限=63。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082019,"parameters":{"m":19,"b":6,"a":3,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"19\",\"6\",\"3\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0020","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-20x+(7)，求 x→4 的去心極限。","answer":{"kind":"short_text","value":"-73","explanation_zh":"去心極限=-73。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082020,"parameters":{"m":-20,"b":7,"a":4,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-20\",\"7\",\"4\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0021","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=21x+(8)，求 x→1 的去心極限。","answer":{"kind":"short_text","value":"29","explanation_zh":"去心極限=29。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082021,"parameters":{"m":21,"b":8,"a":1,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"21\",\"8\",\"1\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0022","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-22x+(9)，求 x→2 的去心極限。","answer":{"kind":"short_text","value":"-35","explanation_zh":"去心極限=-35。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082022,"parameters":{"m":-22,"b":9,"a":2,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-22\",\"9\",\"2\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0023","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=23x+(10)，求 x→3 的去心極限。","answer":{"kind":"short_text","value":"79","explanation_zh":"去心極限=79。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082023,"parameters":{"m":23,"b":10,"a":3,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"23\",\"10\",\"3\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0024","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=-24x+(11)，求 x→4 的去心極限。","answer":{"kind":"short_text","value":"-85","explanation_zh":"去心極限=-85。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082024,"parameters":{"m":-24,"b":11,"a":4,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"-24\",\"11\",\"4\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0025","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射函數的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=25x+(12)，求 x→1 的去心極限。","answer":{"kind":"short_text","value":"37","explanation_zh":"去心極限=37。以明示仿射式的極限規則求值；不把去心極限混同任意未給函數的點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射函數的去心極限"]},"variant":{"family_id":"qf-function-limit-affine","seed":202610082025,"parameters":{"m":25,"b":12,"a":1,"operation":"affine_punctured_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-affine/{\"inputs\":[\"25\",\"12\",\"1\"],\"operation\":\"affine_punctured_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0026","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-12)²)/(x−(-12))，x≠-12。求 x→-12 的極限。","answer":{"kind":"short_text","value":"-24","explanation_zh":"可去商極限=-24。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082026,"parameters":{"a":-12,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-12\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0027","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-11)²)/(x−(-11))，x≠-11。求 x→-11 的極限。","answer":{"kind":"short_text","value":"-22","explanation_zh":"可去商極限=-22。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082027,"parameters":{"a":-11,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-11\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0028","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-10)²)/(x−(-10))，x≠-10。求 x→-10 的極限。","answer":{"kind":"short_text","value":"-20","explanation_zh":"可去商極限=-20。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082028,"parameters":{"a":-10,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-10\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0029","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-9)²)/(x−(-9))，x≠-9。求 x→-9 的極限。","answer":{"kind":"short_text","value":"-18","explanation_zh":"可去商極限=-18。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082029,"parameters":{"a":-9,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-9\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0030","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-8)²)/(x−(-8))，x≠-8。求 x→-8 的極限。","answer":{"kind":"short_text","value":"-16","explanation_zh":"可去商極限=-16。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082030,"parameters":{"a":-8,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-8\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0031","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-7)²)/(x−(-7))，x≠-7。求 x→-7 的極限。","answer":{"kind":"short_text","value":"-14","explanation_zh":"可去商極限=-14。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082031,"parameters":{"a":-7,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-7\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0032","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-6)²)/(x−(-6))，x≠-6。求 x→-6 的極限。","answer":{"kind":"short_text","value":"-12","explanation_zh":"可去商極限=-12。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082032,"parameters":{"a":-6,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-6\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0033","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-5)²)/(x−(-5))，x≠-5。求 x→-5 的極限。","answer":{"kind":"short_text","value":"-10","explanation_zh":"可去商極限=-10。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082033,"parameters":{"a":-5,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-5\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0034","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-4)²)/(x−(-4))，x≠-4。求 x→-4 的極限。","answer":{"kind":"short_text","value":"-8","explanation_zh":"可去商極限=-8。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082034,"parameters":{"a":-4,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-4\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0035","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-3)²)/(x−(-3))，x≠-3。求 x→-3 的極限。","answer":{"kind":"short_text","value":"-6","explanation_zh":"可去商極限=-6。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082035,"parameters":{"a":-3,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-3\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0036","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-2)²)/(x−(-2))，x≠-2。求 x→-2 的極限。","answer":{"kind":"short_text","value":"-4","explanation_zh":"可去商極限=-4。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082036,"parameters":{"a":-2,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-2\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0037","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(-1)²)/(x−(-1))，x≠-1。求 x→-1 的極限。","answer":{"kind":"short_text","value":"-2","explanation_zh":"可去商極限=-2。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082037,"parameters":{"a":-1,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"-1\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0038","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(0)²)/(x−(0))，x≠0。求 x→0 的極限。","answer":{"kind":"short_text","value":"0","explanation_zh":"可去商極限=0。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082038,"parameters":{"a":0,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"0\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0039","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(1)²)/(x−(1))，x≠1。求 x→1 的極限。","answer":{"kind":"short_text","value":"2","explanation_zh":"可去商極限=2。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082039,"parameters":{"a":1,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"1\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0040","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(2)²)/(x−(2))，x≠2。求 x→2 的極限。","answer":{"kind":"short_text","value":"4","explanation_zh":"可去商極限=4。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082040,"parameters":{"a":2,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"2\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0041","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(3)²)/(x−(3))，x≠3。求 x→3 的極限。","answer":{"kind":"short_text","value":"6","explanation_zh":"可去商極限=6。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082041,"parameters":{"a":3,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"3\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0042","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(4)²)/(x−(4))，x≠4。求 x→4 的極限。","answer":{"kind":"short_text","value":"8","explanation_zh":"可去商極限=8。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082042,"parameters":{"a":4,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"4\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0043","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(5)²)/(x−(5))，x≠5。求 x→5 的極限。","answer":{"kind":"short_text","value":"10","explanation_zh":"可去商極限=10。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082043,"parameters":{"a":5,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"5\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0044","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(6)²)/(x−(6))，x≠6。求 x→6 的極限。","answer":{"kind":"short_text","value":"12","explanation_zh":"可去商極限=12。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082044,"parameters":{"a":6,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"6\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0045","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(7)²)/(x−(7))，x≠7。求 x→7 的極限。","answer":{"kind":"short_text","value":"14","explanation_zh":"可去商極限=14。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082045,"parameters":{"a":7,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"7\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0046","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(8)²)/(x−(8))，x≠8。求 x→8 的極限。","answer":{"kind":"short_text","value":"16","explanation_zh":"可去商極限=16。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082046,"parameters":{"a":8,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"8\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0047","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(9)²)/(x−(9))，x≠9。求 x→9 的極限。","answer":{"kind":"short_text","value":"18","explanation_zh":"可去商極限=18。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082047,"parameters":{"a":9,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"9\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0048","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(10)²)/(x−(10))，x≠10。求 x→10 的極限。","answer":{"kind":"short_text","value":"20","explanation_zh":"可去商極限=20。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082048,"parameters":{"a":10,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"10\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0049","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(11)²)/(x−(11))，x≠11。求 x→11 的極限。","answer":{"kind":"short_text","value":"22","explanation_zh":"可去商極限=22。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082049,"parameters":{"a":11,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"11\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0050","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"可去商式的去心極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。f(x)=(x²−(12)²)/(x−(12))，x≠12。求 x→12 的極限。","answer":{"kind":"short_text","value":"24","explanation_zh":"可去商極限=24。去心位置可化為 x+a；原商在 a 未定義，不能代入 0/0 冒充極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可去商式的去心極限"]},"variant":{"family_id":"qf-function-limit-removable-quotient","seed":202610082050,"parameters":{"a":12,"operation":"removable_quotient_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-removable-quotient/{\"inputs\":[\"12\"],\"operation\":\"removable_quotient_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0051","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(0)(x−A)，斜率 m=0。ε=1/1>0、δ=5/1>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(0, 1)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082051,"parameters":{"m":0,"eps_n":1,"eps_d":1,"delta_n":5,"delta_d":1,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"constant_affine\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0052","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-1)(x−A)，斜率 m=-1。ε=2/2>0、δ=2/2>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(4, 4)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082052,"parameters":{"m":-1,"eps_n":2,"eps_d":2,"delta_n":2,"delta_d":2,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"1\",\"1/1\",\"1/1\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0053","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(2)(x−A)，斜率 m=2。ε=3/3>0、δ=3/12>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(18, 36)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082053,"parameters":{"m":2,"eps_n":3,"eps_d":3,"delta_n":3,"delta_d":12,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"2\",\"1/1\",\"1/4\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0054","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-3)(x−A)，斜率 m=-3。ε=1/4>0、δ=2/12>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(24, 12)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082054,"parameters":{"m":-3,"eps_n":1,"eps_d":4,"delta_n":2,"delta_d":12,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"3\",\"1/4\",\"1/6\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0055","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(4)(x−A)，斜率 m=4。ε=2/1>0、δ=2/4>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(8, 8)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082055,"parameters":{"m":4,"eps_n":2,"eps_d":1,"delta_n":2,"delta_d":4,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"4\",\"2/1\",\"1/2\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0056","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-5)(x−A)，斜率 m=-5。ε=3/2>0、δ=3/20>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(30, 60)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082056,"parameters":{"m":-5,"eps_n":3,"eps_d":2,"delta_n":3,"delta_d":20,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"5\",\"3/2\",\"3/20\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0057","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(6)(x−A)，斜率 m=6。ε=1/3>0、δ=2/18>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(36, 18)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082057,"parameters":{"m":6,"eps_n":1,"eps_d":3,"delta_n":2,"delta_d":18,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"6\",\"1/3\",\"1/9\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0058","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-7)(x−A)，斜率 m=-7。ε=2/4>0、δ=2/28>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(56, 56)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082058,"parameters":{"m":-7,"eps_n":2,"eps_d":4,"delta_n":2,"delta_d":28,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"7\",\"1/2\",\"1/14\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0059","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(8)(x−A)，斜率 m=8。ε=3/1>0、δ=3/16>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(24, 48)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082059,"parameters":{"m":8,"eps_n":3,"eps_d":1,"delta_n":3,"delta_d":16,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"8\",\"3/1\",\"3/16\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0060","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-9)(x−A)，斜率 m=-9。ε=1/2>0、δ=2/18>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(36, 18)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082060,"parameters":{"m":-9,"eps_n":1,"eps_d":2,"delta_n":2,"delta_d":18,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"9\",\"1/2\",\"1/9\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0061","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(10)(x−A)，斜率 m=10。ε=2/3>0、δ=2/30>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(60, 60)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082061,"parameters":{"m":10,"eps_n":2,"eps_d":3,"delta_n":2,"delta_d":30,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"10\",\"2/3\",\"1/15\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0062","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-11)(x−A)，斜率 m=-11。ε=3/4>0、δ=3/88>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(132, 264)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082062,"parameters":{"m":-11,"eps_n":3,"eps_d":4,"delta_n":3,"delta_d":88,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"11\",\"3/4\",\"3/88\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0063","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(12)(x−A)，斜率 m=12。ε=1/1>0、δ=2/12>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(24, 12)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082063,"parameters":{"m":12,"eps_n":1,"eps_d":1,"delta_n":2,"delta_d":12,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"12\",\"1/1\",\"1/6\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0064","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-13)(x−A)，斜率 m=-13。ε=2/2>0、δ=2/26>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(52, 52)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082064,"parameters":{"m":-13,"eps_n":2,"eps_d":2,"delta_n":2,"delta_d":26,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"13\",\"1/1\",\"1/13\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0065","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(14)(x−A)，斜率 m=14。ε=3/3>0、δ=3/84>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(126, 252)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082065,"parameters":{"m":14,"eps_n":3,"eps_d":3,"delta_n":3,"delta_d":84,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"14\",\"1/1\",\"1/28\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0066","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-15)(x−A)，斜率 m=-15。ε=1/4>0、δ=2/60>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(120, 60)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082066,"parameters":{"m":-15,"eps_n":1,"eps_d":4,"delta_n":2,"delta_d":60,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"15\",\"1/4\",\"1/30\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0067","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(16)(x−A)，斜率 m=16。ε=2/1>0、δ=2/16>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(32, 32)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082067,"parameters":{"m":16,"eps_n":2,"eps_d":1,"delta_n":2,"delta_d":16,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"16\",\"2/1\",\"1/8\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0068","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-17)(x−A)，斜率 m=-17。ε=3/2>0、δ=3/68>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(102, 204)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082068,"parameters":{"m":-17,"eps_n":3,"eps_d":2,"delta_n":3,"delta_d":68,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"17\",\"3/2\",\"3/68\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0069","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(18)(x−A)，斜率 m=18。ε=1/3>0、δ=2/54>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(108, 54)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082069,"parameters":{"m":18,"eps_n":1,"eps_d":3,"delta_n":2,"delta_d":54,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"18\",\"1/3\",\"1/27\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0070","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-19)(x−A)，斜率 m=-19。ε=2/4>0、δ=2/76>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(152, 152)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082070,"parameters":{"m":-19,"eps_n":2,"eps_d":4,"delta_n":2,"delta_d":76,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"19\",\"1/2\",\"1/38\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0071","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(20)(x−A)，斜率 m=20。ε=3/1>0、δ=3/40>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(60, 120)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082071,"parameters":{"m":20,"eps_n":3,"eps_d":1,"delta_n":3,"delta_d":40,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"20\",\"3/1\",\"3/40\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0072","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-21)(x−A)，斜率 m=-21。ε=1/2>0、δ=2/42>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(84, 42)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082072,"parameters":{"m":-21,"eps_n":1,"eps_d":2,"delta_n":2,"delta_d":42,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"21\",\"1/2\",\"1/21\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0073","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(22)(x−A)，斜率 m=22。ε=2/3>0、δ=2/66>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(132, 132)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082073,"parameters":{"m":22,"eps_n":2,"eps_d":3,"delta_n":2,"delta_d":66,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"22\",\"2/3\",\"1/33\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0074","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(-23)(x−A)，斜率 m=-23。ε=3/4>0、δ=3/184>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"仿射界交叉乘積=(276, 552)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082074,"parameters":{"m":-23,"eps_n":3,"eps_d":4,"delta_n":3,"delta_d":184,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"23\",\"3/4\",\"3/184\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0075","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"仿射 δ 見證的嚴格內部","stem_zh":"去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(24)(x−A)，斜率 m=24。ε=1/1>0、δ=2/24>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"仿射界交叉乘積=(48, 24)；非嚴格比較=否。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射 δ 見證的嚴格內部"]},"variant":{"family_id":"qf-function-limit-affine-delta","seed":202610082075,"parameters":{"m":24,"eps_n":1,"eps_d":1,"delta_n":2,"delta_d":24,"operation":"affine_delta_validity","domain":"punctured_reals","semantic_template":"qf-function-limit-affine-delta/{\"inputs\":[\"24\",\"1/1\",\"1/12\"],\"operation\":\"affine_delta_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0076","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-12 時 f(x)=1x+(0)，x>-12 時 f(x)=-1x+(-24)。求 x→-12 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"-12","explanation_zh":"左右極限=(-12, -12)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082076,"parameters":{"a":-12,"left_m":1,"left_b":0,"right_m":-1,"right_b":-24,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-12\",\"-12\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0077","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-11 時 f(x)=2x+(11)，x>-11 時 f(x)=-2x+(-20)。求 x→-11 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-11, 2)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082077,"parameters":{"a":-11,"left_m":2,"left_b":11,"right_m":-2,"right_b":-20,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-11\",\"2\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0078","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-10 時 f(x)=3x+(20)，x>-10 時 f(x)=-3x+(-27)。求 x→-10 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-10, 3)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082078,"parameters":{"a":-10,"left_m":3,"left_b":20,"right_m":-3,"right_b":-27,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-10\",\"3\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0079","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-9 時 f(x)=4x+(27)，x>-9 時 f(x)=-1x+(-18)。求 x→-9 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"-9","explanation_zh":"左右極限=(-9, -9)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082079,"parameters":{"a":-9,"left_m":4,"left_b":27,"right_m":-1,"right_b":-18,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-9\",\"-9\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0080","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-8 時 f(x)=1x+(0)，x>-8 時 f(x)=-2x+(-11)。求 x→-8 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-8, 5)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082080,"parameters":{"a":-8,"left_m":1,"left_b":0,"right_m":-2,"right_b":-11,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-8\",\"5\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0081","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-7 時 f(x)=2x+(7)，x>-7 時 f(x)=-3x+(-15)。求 x→-7 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-7, 6)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082081,"parameters":{"a":-7,"left_m":2,"left_b":7,"right_m":-3,"right_b":-15,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-7\",\"6\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0082","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-6 時 f(x)=3x+(12)，x>-6 時 f(x)=-1x+(-12)。求 x→-6 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"-6","explanation_zh":"左右極限=(-6, -6)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082082,"parameters":{"a":-6,"left_m":3,"left_b":12,"right_m":-1,"right_b":-12,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-6\",\"-6\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0083","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-5 時 f(x)=4x+(15)，x>-5 時 f(x)=-2x+(-2)。求 x→-5 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-5, 8)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082083,"parameters":{"a":-5,"left_m":4,"left_b":15,"right_m":-2,"right_b":-2,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-5\",\"8\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0084","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-4 時 f(x)=1x+(0)，x>-4 時 f(x)=-3x+(-3)。求 x→-4 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-4, 9)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082084,"parameters":{"a":-4,"left_m":1,"left_b":0,"right_m":-3,"right_b":-3,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-4\",\"9\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0085","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-3 時 f(x)=2x+(3)，x>-3 時 f(x)=-1x+(-6)。求 x→-3 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"-3","explanation_zh":"左右極限=(-3, -3)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082085,"parameters":{"a":-3,"left_m":2,"left_b":3,"right_m":-1,"right_b":-6,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-3\",\"-3\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0086","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-2 時 f(x)=3x+(4)，x>-2 時 f(x)=-2x+(7)。求 x→-2 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-2, 11)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082086,"parameters":{"a":-2,"left_m":3,"left_b":4,"right_m":-2,"right_b":7,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-2\",\"11\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0087","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<-1 時 f(x)=4x+(3)，x>-1 時 f(x)=-3x+(9)。求 x→-1 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(-1, 12)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082087,"parameters":{"a":-1,"left_m":4,"left_b":3,"right_m":-3,"right_b":9,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"-1\",\"12\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0088","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<0 時 f(x)=1x+(0)，x>0 時 f(x)=-1x+(0)。求 x→0 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"0","explanation_zh":"左右極限=(0, 0)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082088,"parameters":{"a":0,"left_m":1,"left_b":0,"right_m":-1,"right_b":0,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"0\",\"0\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0089","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<1 時 f(x)=2x+(-1)，x>1 時 f(x)=-2x+(16)。求 x→1 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(1, 14)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082089,"parameters":{"a":1,"left_m":2,"left_b":-1,"right_m":-2,"right_b":16,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"1\",\"14\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0090","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<2 時 f(x)=3x+(-4)，x>2 時 f(x)=-3x+(21)。求 x→2 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(2, 15)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082090,"parameters":{"a":2,"left_m":3,"left_b":-4,"right_m":-3,"right_b":21,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"2\",\"15\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0091","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<3 時 f(x)=4x+(-9)，x>3 時 f(x)=-1x+(6)。求 x→3 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"3","explanation_zh":"左右極限=(3, 3)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082091,"parameters":{"a":3,"left_m":4,"left_b":-9,"right_m":-1,"right_b":6,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"3\",\"3\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0092","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<4 時 f(x)=1x+(0)，x>4 時 f(x)=-2x+(25)。求 x→4 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(4, 17)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082092,"parameters":{"a":4,"left_m":1,"left_b":0,"right_m":-2,"right_b":25,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"4\",\"17\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0093","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<5 時 f(x)=2x+(-5)，x>5 時 f(x)=-3x+(33)。求 x→5 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(5, 18)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082093,"parameters":{"a":5,"left_m":2,"left_b":-5,"right_m":-3,"right_b":33,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"5\",\"18\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0094","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<6 時 f(x)=3x+(-12)，x>6 時 f(x)=-1x+(12)。求 x→6 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"6","explanation_zh":"左右極限=(6, 6)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082094,"parameters":{"a":6,"left_m":3,"left_b":-12,"right_m":-1,"right_b":12,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"6\",\"6\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0095","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<7 時 f(x)=4x+(-21)，x>7 時 f(x)=-2x+(34)。求 x→7 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(7, 20)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082095,"parameters":{"a":7,"left_m":4,"left_b":-21,"right_m":-2,"right_b":34,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"7\",\"20\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0096","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<8 時 f(x)=1x+(0)，x>8 時 f(x)=-3x+(45)。求 x→8 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(8, 21)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082096,"parameters":{"a":8,"left_m":1,"left_b":0,"right_m":-3,"right_b":45,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"8\",\"21\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0097","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<9 時 f(x)=2x+(-9)，x>9 時 f(x)=-1x+(18)。求 x→9 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"9","explanation_zh":"左右極限=(9, 9)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082097,"parameters":{"a":9,"left_m":2,"left_b":-9,"right_m":-1,"right_b":18,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"9\",\"9\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0098","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<10 時 f(x)=3x+(-20)，x>10 時 f(x)=-2x+(43)。求 x→10 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(10, 23)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082098,"parameters":{"a":10,"left_m":3,"left_b":-20,"right_m":-2,"right_b":43,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"10\",\"23\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0099","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<11 時 f(x)=4x+(-33)，x>11 時 f(x)=-3x+(57)。求 x→11 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"左右極限=(11, 24)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082099,"parameters":{"a":11,"left_m":4,"left_b":-33,"right_m":-3,"right_b":57,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"11\",\"24\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r02-function-limit-0100","batch_id":"batch-20261007-advance-20261008-r02-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"兩仿射支的雙側極限","stem_zh":"去心實數論域，採完整兩側實數鄰域。當 x<12 時 f(x)=1x+(0)，x>12 時 f(x)=-1x+(24)。求 x→12 的雙側極限，左右不一致答「不存在」；f(a) 不參與此題。","answer":{"kind":"short_text","value":"12","explanation_zh":"左右極限=(12, 12)。先求各側的仿射極限，再比較兩個極限值；去心資料不自行補定點值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩仿射支的雙側極限"]},"variant":{"family_id":"qf-function-limit-two-sided-affine","seed":202610082100,"parameters":{"a":12,"left_m":1,"left_b":0,"right_m":-1,"right_b":24,"operation":"two_sided_affine_limit","domain":"punctured_reals","semantic_template":"qf-function-limit-two-sided-affine/{\"inputs\":[\"12\",\"12\"],\"operation\":\"two_sided_affine_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}