{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r03-function-limit-0001","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"失敗δ的精確去心反例","stem_zh":"實數模型；點值與去心極限分別解讀。固定實數A、L，x≠A時 f(x)=L+(1)(x−A)。ε=1/1、δ=2/1 且 |m|δ>ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0ε。取 h=ε/|m|，列 (h, |f(A+h)−L|)，使0a時 f(x)=-1x+(-14)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-12, -12)","explanation_zh":"左右極限=(-12, -12)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093051,"parameters":{"a":-2,"left_m":1,"left_b":-10,"right_m":-1,"right_b":-14,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-12\",\"-12\"]"}},"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-20261009-r03-function-limit-0052","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1；xa時 f(x)=-2x+(10)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-11, 12)","explanation_zh":"左右極限=(-11, 12)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093052,"parameters":{"a":-1,"left_m":2,"left_b":-9,"right_m":-2,"right_b":10,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-11\",\"12\"]"}},"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-20261009-r03-function-limit-0053","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=0；xa時 f(x)=-1x+(11)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-10, 11)","explanation_zh":"左右極限=(-10, 11)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093053,"parameters":{"a":0,"left_m":3,"left_b":-10,"right_m":-1,"right_b":11,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-10\",\"11\"]"}},"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-20261009-r03-function-limit-0054","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=1；xa時 f(x)=-2x+(-7)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-9, -9)","explanation_zh":"左右極限=(-9, -9)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093054,"parameters":{"a":1,"left_m":1,"left_b":-10,"right_m":-2,"right_b":-7,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-9\",\"-9\"]"}},"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-20261009-r03-function-limit-0055","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2；xa時 f(x)=-1x+(7)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-8, 9)","explanation_zh":"左右極限=(-8, 9)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093055,"parameters":{"a":-2,"left_m":2,"left_b":-4,"right_m":-1,"right_b":7,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-8\",\"9\"]"}},"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-20261009-r03-function-limit-0056","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1；xa時 f(x)=-2x+(6)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-7, 8)","explanation_zh":"左右極限=(-7, 8)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093056,"parameters":{"a":-1,"left_m":3,"left_b":-4,"right_m":-2,"right_b":6,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-7\",\"8\"]"}},"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-20261009-r03-function-limit-0057","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=0；xa時 f(x)=-1x+(-6)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-6, -6)","explanation_zh":"左右極限=(-6, -6)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093057,"parameters":{"a":0,"left_m":1,"left_b":-6,"right_m":-1,"right_b":-6,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-6\",\"-6\"]"}},"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-20261009-r03-function-limit-0058","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=1；xa時 f(x)=-2x+(8)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-5, 6)","explanation_zh":"左右極限=(-5, 6)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093058,"parameters":{"a":1,"left_m":2,"left_b":-7,"right_m":-2,"right_b":8,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-5\",\"6\"]"}},"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-20261009-r03-function-limit-0059","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2；xa時 f(x)=-1x+(3)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-4, 5)","explanation_zh":"左右極限=(-4, 5)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093059,"parameters":{"a":-2,"left_m":3,"left_b":2,"right_m":-1,"right_b":3,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-4\",\"5\"]"}},"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-20261009-r03-function-limit-0060","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1；xa時 f(x)=-2x+(-5)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-3, -3)","explanation_zh":"左右極限=(-3, -3)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093060,"parameters":{"a":-1,"left_m":1,"left_b":-2,"right_m":-2,"right_b":-5,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-3\",\"-3\"]"}},"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-20261009-r03-function-limit-0061","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=0；xa時 f(x)=-1x+(3)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-2, 3)","explanation_zh":"左右極限=(-2, 3)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093061,"parameters":{"a":0,"left_m":2,"left_b":-2,"right_m":-1,"right_b":3,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-2\",\"3\"]"}},"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-20261009-r03-function-limit-0062","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=1；xa時 f(x)=-2x+(4)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(-1, 2)","explanation_zh":"左右極限=(-1, 2)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093062,"parameters":{"a":1,"left_m":3,"left_b":-4,"right_m":-2,"right_b":4,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"-1\",\"2\"]"}},"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-20261009-r03-function-limit-0063","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2；xa時 f(x)=-1x+(-2)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"左右極限=(0, 0)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093063,"parameters":{"a":-2,"left_m":1,"left_b":2,"right_m":-1,"right_b":-2,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"0\",\"0\"]"}},"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-20261009-r03-function-limit-0064","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1；xa時 f(x)=-2x+(-2)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(1, 0)","explanation_zh":"左右極限=(1, 0)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093064,"parameters":{"a":-1,"left_m":2,"left_b":3,"right_m":-2,"right_b":-2,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"1\",\"0\"]"}},"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-20261009-r03-function-limit-0065","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=0；xa時 f(x)=-1x+(-1)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(2, -1)","explanation_zh":"左右極限=(2, -1)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093065,"parameters":{"a":0,"left_m":3,"left_b":2,"right_m":-1,"right_b":-1,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"2\",\"-1\"]"}},"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-20261009-r03-function-limit-0066","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=1；xa時 f(x)=-2x+(5)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(3, 3)","explanation_zh":"左右極限=(3, 3)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093066,"parameters":{"a":1,"left_m":1,"left_b":2,"right_m":-2,"right_b":5,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"3\",\"3\"]"}},"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-20261009-r03-function-limit-0067","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2；xa時 f(x)=-1x+(-5)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(4, -3)","explanation_zh":"左右極限=(4, -3)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093067,"parameters":{"a":-2,"left_m":2,"left_b":8,"right_m":-1,"right_b":-5,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"4\",\"-3\"]"}},"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-20261009-r03-function-limit-0068","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1；xa時 f(x)=-2x+(-6)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(5, -4)","explanation_zh":"左右極限=(5, -4)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093068,"parameters":{"a":-1,"left_m":3,"left_b":8,"right_m":-2,"right_b":-6,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"5\",\"-4\"]"}},"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-20261009-r03-function-limit-0069","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=0；xa時 f(x)=-1x+(6)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(6, 6)","explanation_zh":"左右極限=(6, 6)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093069,"parameters":{"a":0,"left_m":1,"left_b":6,"right_m":-1,"right_b":6,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"6\",\"6\"]"}},"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-20261009-r03-function-limit-0070","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=1；xa時 f(x)=-2x+(-4)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(7, -6)","explanation_zh":"左右極限=(7, -6)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093070,"parameters":{"a":1,"left_m":2,"left_b":5,"right_m":-2,"right_b":-4,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"7\",\"-6\"]"}},"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-20261009-r03-function-limit-0071","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2；xa時 f(x)=-1x+(-9)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(8, -7)","explanation_zh":"左右極限=(8, -7)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093071,"parameters":{"a":-2,"left_m":3,"left_b":14,"right_m":-1,"right_b":-9,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"8\",\"-7\"]"}},"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-20261009-r03-function-limit-0072","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1；xa時 f(x)=-2x+(7)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(9, 9)","explanation_zh":"左右極限=(9, 9)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093072,"parameters":{"a":-1,"left_m":1,"left_b":10,"right_m":-2,"right_b":7,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"9\",\"9\"]"}},"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-20261009-r03-function-limit-0073","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=0；xa時 f(x)=-1x+(-9)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(10, -9)","explanation_zh":"左右極限=(10, -9)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093073,"parameters":{"a":0,"left_m":2,"left_b":10,"right_m":-1,"right_b":-9,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"10\",\"-9\"]"}},"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-20261009-r03-function-limit-0074","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=1；xa時 f(x)=-2x+(-8)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(11, -10)","explanation_zh":"左右極限=(11, -10)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093074,"parameters":{"a":1,"left_m":3,"left_b":8,"right_m":-2,"right_b":-8,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"11\",\"-10\"]"}},"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-20261009-r03-function-limit-0075","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"左右去心極限的有序對","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2；xa時 f(x)=-1x+(10)。列 (左極限, 右極限)，不交換順序。","answer":{"kind":"expression","value":"(12, 12)","explanation_zh":"左右極限=(12, 12)。各支分別求極限；兩側相同才有相同雙側候選，此題不補定中心值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["左右去心極限的有序對"]},"variant":{"family_id":"qf-density-limit-one-sided-pair","seed":202610093075,"parameters":{"a":-2,"left_m":1,"left_b":14,"right_m":-1,"right_b":10,"operation":"density_limit_one_sided_pair","domain":"punctured_reals","semantic_template":"qf-density-limit-one-sided-pair/[\"12\",\"12\"]"}},"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-20261009-r03-function-limit-0076","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-12，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-11 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-23/1","explanation_zh":"原商合法點值=-23/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093076,"parameters":{"a":-12,"x":-11,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-12,-11]"}},"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-20261009-r03-function-limit-0077","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-11，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-13 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-24/1","explanation_zh":"原商合法點值=-24/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093077,"parameters":{"a":-11,"x":-13,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-11,-13]"}},"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-20261009-r03-function-limit-0078","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-10，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-7 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-17/1","explanation_zh":"原商合法點值=-17/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093078,"parameters":{"a":-10,"x":-7,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-10,-7]"}},"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-20261009-r03-function-limit-0079","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-9，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-13 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-22/1","explanation_zh":"原商合法點值=-22/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093079,"parameters":{"a":-9,"x":-13,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-9,-13]"}},"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-20261009-r03-function-limit-0080","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-8，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-7 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-15/1","explanation_zh":"原商合法點值=-15/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093080,"parameters":{"a":-8,"x":-7,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-8,-7]"}},"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-20261009-r03-function-limit-0081","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-7，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-9 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-16/1","explanation_zh":"原商合法點值=-16/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093081,"parameters":{"a":-7,"x":-9,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-7,-9]"}},"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-20261009-r03-function-limit-0082","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-6，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-3 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"原商合法點值=-9/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093082,"parameters":{"a":-6,"x":-3,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-6,-3]"}},"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-20261009-r03-function-limit-0083","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-5，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-9 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-14/1","explanation_zh":"原商合法點值=-14/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093083,"parameters":{"a":-5,"x":-9,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-5,-9]"}},"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-20261009-r03-function-limit-0084","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-4，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-3 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-7/1","explanation_zh":"原商合法點值=-7/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093084,"parameters":{"a":-4,"x":-3,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-4,-3]"}},"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-20261009-r03-function-limit-0085","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-3，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-5 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"原商合法點值=-8/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093085,"parameters":{"a":-3,"x":-5,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-3,-5]"}},"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-20261009-r03-function-limit-0086","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-2，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=1 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"原商合法點值=-1/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093086,"parameters":{"a":-2,"x":1,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-2,1]"}},"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-20261009-r03-function-limit-0087","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=-1，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-5 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"-6/1","explanation_zh":"原商合法點值=-6/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093087,"parameters":{"a":-1,"x":-5,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[-1,-5]"}},"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-20261009-r03-function-limit-0088","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=0，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=1 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"1/1","explanation_zh":"原商合法點值=1/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093088,"parameters":{"a":0,"x":1,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[0,1]"}},"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-20261009-r03-function-limit-0089","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=1，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-1 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"0/1","explanation_zh":"原商合法點值=0/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093089,"parameters":{"a":1,"x":-1,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[1,-1]"}},"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-20261009-r03-function-limit-0090","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=2，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=5 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"7/1","explanation_zh":"原商合法點值=7/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093090,"parameters":{"a":2,"x":5,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[2,5]"}},"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-20261009-r03-function-limit-0091","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=3，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=-1 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"2/1","explanation_zh":"原商合法點值=2/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093091,"parameters":{"a":3,"x":-1,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[3,-1]"}},"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-20261009-r03-function-limit-0092","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=4，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=5 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"9/1","explanation_zh":"原商合法點值=9/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093092,"parameters":{"a":4,"x":5,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[4,5]"}},"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-20261009-r03-function-limit-0093","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=5，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=3 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"8/1","explanation_zh":"原商合法點值=8/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093093,"parameters":{"a":5,"x":3,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[5,3]"}},"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-20261009-r03-function-limit-0094","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=6，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=9 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"15/1","explanation_zh":"原商合法點值=15/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093094,"parameters":{"a":6,"x":9,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[6,9]"}},"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-20261009-r03-function-limit-0095","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=7，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=3 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"10/1","explanation_zh":"原商合法點值=10/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093095,"parameters":{"a":7,"x":3,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[7,3]"}},"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-20261009-r03-function-limit-0096","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=8，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=9 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"17/1","explanation_zh":"原商合法點值=17/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093096,"parameters":{"a":8,"x":9,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[8,9]"}},"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-20261009-r03-function-limit-0097","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=9，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=7 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"16/1","explanation_zh":"原商合法點值=16/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093097,"parameters":{"a":9,"x":7,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[9,7]"}},"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-20261009-r03-function-limit-0098","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=10，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=13 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"23/1","explanation_zh":"原商合法點值=23/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093098,"parameters":{"a":10,"x":13,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[10,13]"}},"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-20261009-r03-function-limit-0099","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=11，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=7 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"18/1","explanation_zh":"原商合法點值=18/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093099,"parameters":{"a":11,"x":7,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[11,7]"}},"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-20261009-r03-function-limit-0100","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-function-limit"],"title_zh":"原商在合法去心點的值","stem_zh":"實數模型；點值與去心極限分別解讀。a=12，原商 f(x)=(x²−a²)/(x−a) 僅在 x≠a 定義。求 x=13 的值，採最簡正分母分數；這不是未定義中心點值或去心極限的替代。","answer":{"kind":"expression","value":"25/1","explanation_zh":"原商合法點值=25/1。本次指定點分母非零，直接以整數平方差相除；未填原商在a的值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["原商在合法去心點的值"]},"variant":{"family_id":"qf-density-limit-punctured-quotient-value","seed":202610093100,"parameters":{"a":12,"x":13,"operation":"density_limit_punctured_quotient_value","domain":"punctured_reals","semantic_template":"qf-density-limit-punctured-quotient-value/[12,13]"}},"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"}}