{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261008-r02-function-limit-0089", "pilot_number": "R0063-32339", "number": 32339, "family_id": "qf-function-limit-two-sided-affine", "title": { "zh-Hant": "第 32339 題：兩仿射支的雙側極限", "en": "Question 32339: Two-Sided Limit of Affine Branches" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261007-advance-20261008-r02-function-limit-0089", "batch_id": "batch-20261007-advance-20261008-r02-function-limit-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b4c42da070a641d60674af95d2666bbfa11760ffcfb2f490640c9f2c0dc1e406", "family_id": "qf-function-limit-two-sided-affine", "seed": 202610082089, "semantic_template": "qf-function-limit-two-sided-affine/{\"inputs\":[\"1\",\"14\"],\"operation\":\"two_sided_affine_limit\"}" }, "fields": { "title": "Two-Sided Limit of Affine Branches", "stem": "On the punctured domain R∖{1}, The left branch at x<1 is (2)x+(-1); the right branch at x>1 is (-2)x+(16). Find the finite two-sided limit, or answer does not exist when the two side limits differ.", "answer": { "kind": "short_text", "value": "does not exist", "explanation": "The left and right limits are [1, 14]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit." }, "skills": [ "Two-Sided Limit of Affine Branches" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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。", "en": "On the punctured domain R∖{1}, The left branch at x<1 is (2)x+(-1); the right branch at x>1 is (-2)x+(16). Find the finite two-sided limit, or answer does not exist when the two side limits differ." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-function-limit-two-sided-affine, two_sided_affine_limit, punctured_reals, []]", "after": "derived = [[integer_approach_value_OR_source_nonexistence_STRING]]", "relation": "implies", "rule_id": "R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle", "conditions": [], "explanation": { "zh-Hant": "BASE來源沒有本輪legacy的通用operation/domain/額外欄位禁令；保metadata而不當答案。PULSE與legacy則核來源精確欄位、operation及domain。不要把tuple、/1或輸入界強套全部題族。", "en": "BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "LEFT affine approach ORIGINAL slope times approach center := mul(2, 1)", "after": "LEFT affine approach ORIGINAL slope times approach center = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 LEFT affine approach ORIGINAL slope times approach center，不以小數近似替代。", "en": "Recompute LEFT affine approach ORIGINAL slope times approach center with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "LEFT affine approach ORIGINAL slope times approach center" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "LEFT affine approach ORIGINAL approach affine constant sum := sum(2, -1)", "after": "LEFT affine approach ORIGINAL approach affine constant sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 LEFT affine approach ORIGINAL approach affine constant sum，不以小數近似替代。", "en": "Recompute LEFT affine approach ORIGINAL approach affine constant sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "-1" ], "output": "1", "label": "LEFT affine approach ORIGINAL approach affine constant sum" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "LEFT affine approach ORIGINAL positive slope bound one plus abs m := sum(1, 2)", "after": "LEFT affine approach ORIGINAL positive slope bound one plus abs m = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 LEFT affine approach ORIGINAL positive slope bound one plus abs m，不以小數近似替代。", "en": "Recompute LEFT affine approach ORIGINAL positive slope bound one plus abs m with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "2" ], "output": "3", "label": "LEFT affine approach ORIGINAL positive slope bound one plus abs m" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [LEFT affine approach, 2, -1, 1]", "after": "derived = [1, 3, error = m*(x-a), delta = epsilon/3]", "relation": "implies", "rule_id": "Limit_AFFINE_whole_law_error_epsilon_delta_NOT_center_assignment", "conditions": [], "explanation": { "zh-Hant": "整個已給仿射law滿足f(x)−L=m(x−a)。任意ε>0取δ=ε/(1+|m|)>0，在該側／去心域0<|x−a|<δ使誤差<ε；m0時誤差0也成立。L是approach值，不替原函數定義f(a)，不是抽樣證明。", "en": "For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "RIGHT affine approach ORIGINAL slope times approach center := mul(-2, 1)", "after": "RIGHT affine approach ORIGINAL slope times approach center = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 RIGHT affine approach ORIGINAL slope times approach center，不以小數近似替代。", "en": "Recompute RIGHT affine approach ORIGINAL slope times approach center with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "RIGHT affine approach ORIGINAL slope times approach center" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "RIGHT affine approach ORIGINAL approach affine constant sum := sum(-2, 16)", "after": "RIGHT affine approach ORIGINAL approach affine constant sum = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 RIGHT affine approach ORIGINAL approach affine constant sum，不以小數近似替代。", "en": "Recompute RIGHT affine approach ORIGINAL approach affine constant sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "16" ], "output": "14", "label": "RIGHT affine approach ORIGINAL approach affine constant sum" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "RIGHT affine approach ORIGINAL positive slope bound one plus abs m := sum(1, 2)", "after": "RIGHT affine approach ORIGINAL positive slope bound one plus abs m = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 RIGHT affine approach ORIGINAL positive slope bound one plus abs m，不以小數近似替代。", "en": "Recompute RIGHT affine approach ORIGINAL positive slope bound one plus abs m with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "2" ], "output": "3", "label": "RIGHT affine approach ORIGINAL positive slope bound one plus abs m" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [RIGHT affine approach, -2, 16, 1]", "after": "derived = [14, 3, error = m*(x-a), delta = epsilon/3]", "relation": "implies", "rule_id": "Limit_AFFINE_whole_law_error_epsilon_delta_NOT_center_assignment", "conditions": [], "explanation": { "zh-Hant": "整個已給仿射law滿足f(x)−L=m(x−a)。任意ε>0取δ=ε/(1+|m|)>0，在該側／去心域0<|x−a|<δ使誤差<ε；m0時誤差0也成立。L是approach值，不替原函數定義f(a)，不是抽樣證明。", "en": "For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Two-sided ORIGINAL left minus right approach values := sub(1, 14)", "after": "Two-sided ORIGINAL left minus right approach values = -13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Two-sided ORIGINAL left minus right approach values，不以小數近似替代。", "en": "Recompute Two-sided ORIGINAL left minus right approach values with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "14" ], "output": "-13", "label": "Two-sided ORIGINAL left minus right approach values" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Two-sided ORIGINAL positive separation scale gap over3 := div(13, 3)", "after": "Two-sided ORIGINAL positive separation scale gap over3 = 13/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Two-sided ORIGINAL positive separation scale gap over3，不以小數近似替代。", "en": "Recompute Two-sided ORIGINAL positive separation scale gap over3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "3" ], "output": "13/3", "label": "Two-sided ORIGINAL positive separation scale gap over3" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [1, 1, 14]", "after": "derived = [-13, 13/3, two-sided-limit: 不存在]", "relation": "implies", "rule_id": "Limit_TWO_sides_DIFFER_necessity_uniqueness_context_qualified_absence", "conditions": [], "explanation": { "zh-Hant": "若雙側極限存在，其左右限制都必有同一極限；實數極限唯一，但已證兩側值不同，故不可能。正距離/3可分離這兩個值的鄰域。不存在的是共同雙側極限，不刪除兩個單側極限，也不憑抽樣判斷。", "en": "If a two-sided limit existed,BOTH side restrictions would have that SAME limit.Real limits are unique,but the proven side values differ,so this is impossible.Positive gap/3 separates their neighborhoods.What fails is the COMMON two-sided limit,not either one-sided limit;this is not a sample-based decision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "不存在" }, "conclusion": { "canonical_value": "不存在", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261007-advance-20261008-r02-function-limit-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r02-function-limit-foundations/questions.jsonl", "batch_sha256": "1176f30642e50d59427c3026d10f3097c800f4e5fa9dd6f7e3796b1c401da318", "original_line": 89, "original_line_sha256": "1197c41a5ad981d62d3c40eae869312066f9ed3f2be51af161195d95b45375fa", "question_compact_sha256": "b4c42da070a641d60674af95d2666bbfa11760ffcfb2f490640c9f2c0dc1e406", "english_row_compact_sha256": "3d3ae38009dcf1426a223b24a69944c6a4815d0e419cd4892093f202bce795d5", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r02-function-limit-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r02-function-limit-foundations.en.jsonl" }, "mko_bindings": { "mko-function-limit": { "sha256": "2dce49b3fcec9f9cf28bc85107ed2884a57b36ccc297c60e4ef5836b4a3c5ca3", "english_sha256": "608f319ceb9afeb2eb284c43c4a1d6d8d19c4e40dce81f0ac4df60cbb55f4c44", "original_path": "public/data/mko/function-limit.json", "english_path": "public/data/i18n/editions/mko-function-limit.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "f3271d6047df957dbf3d61120bd3dd374784c828b17849887e14caf59d93f745", "original_candidate_sha256": "f7922277b354e04cf0e6e196af85cf2a068114507d148b5f16873946137bfd3b", "original_candidate_line_sha256": "f7922277b354e04cf0e6e196af85cf2a068114507d148b5f16873946137bfd3b" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-011-p64", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0", "question_first_witness_sha256": "eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "b54b15bce9ff7080d029a482f8974fa3a6282609d491087f4a49ace8b1f93e44", "arithmetic_steps": 8 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }