{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0001","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4048ad18403fa18d220469ec277d88559b674969691eaf47636fcea5bf8e39c7","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093001,"semantic_template":"qf-density-limit-invalid-delta-witness/[1,\"1/1\",\"2/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(1)(x−A) for x≠A. Given ε=1/1>0 and failed δ=2/1>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation":"The witness pair is (1/1, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0002","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cff439d3cadd3f74881319091062000266d3f8fa8bb43122dc4f98cb4e7693b4","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093002,"semantic_template":"qf-density-limit-invalid-delta-witness/[2,\"1/1\",\"3/2\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-2)(x−A) for x≠A. Given ε=2/2>0 and failed δ=6/4>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 1/1)","explanation":"The witness pair is (1/2, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0003","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"602676d5da9d319d259854774c6fcc52ca6fb6fe5cbbbf1518ecab6bfa95d008","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093003,"semantic_template":"qf-density-limit-invalid-delta-witness/[3,\"1/1\",\"4/3\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(3)(x−A) for x≠A. Given ε=3/3>0 and failed δ=12/9>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 1/1)","explanation":"The witness pair is (1/3, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0004","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b804149cc0c5d70060f48a67d7a7029225fa1b57b71b14612d2c718d382e9244","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093004,"semantic_template":"qf-density-limit-invalid-delta-witness/[4,\"1/1\",\"1/2\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-4)(x−A) for x≠A. Given ε=4/4>0 and failed δ=8/16>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/4, 1/1)","explanation":"The witness pair is (1/4, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0005","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"511e82c6228b2c91843eb4deb2cee11ca45aa391a9535cf843f3974ae0aa2016","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093005,"semantic_template":"qf-density-limit-invalid-delta-witness/[5,\"1/1\",\"3/5\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(5)(x−A) for x≠A. Given ε=5/5>0 and failed δ=15/25>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/5, 1/1)","explanation":"The witness pair is (1/5, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0006","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bea0aa4ad25c4c70d62799595e383e7d3cbf9d648b473a298d2d0071b5c3559","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093006,"semantic_template":"qf-density-limit-invalid-delta-witness/[6,\"6/1\",\"4/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-6)(x−A) for x≠A. Given ε=6/1>0 and failed δ=24/6>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 6/1)","explanation":"The witness pair is (1/1, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0007","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"419c4719f19f722c17d1b4b539862dc6cb7a76af823c1ea1c2fe8879e4a9bced","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093007,"semantic_template":"qf-density-limit-invalid-delta-witness/[7,\"7/2\",\"1/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(7)(x−A) for x≠A. Given ε=7/2>0 and failed δ=14/14>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 7/2)","explanation":"The witness pair is (1/2, 7/2): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0008","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db62158fb57934349d0b74216bd4975cef484239e064a5f013b68fc2ae23d685","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093008,"semantic_template":"qf-density-limit-invalid-delta-witness/[8,\"8/3\",\"1/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-8)(x−A) for x≠A. Given ε=8/3>0 and failed δ=24/24>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 8/3)","explanation":"The witness pair is (1/3, 8/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0009","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97ba4e40bfc4578c2b6716990d5f02a53a3b628b159e74b175a9d87c4ef04a09","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093009,"semantic_template":"qf-density-limit-invalid-delta-witness/[9,\"9/4\",\"1/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(9)(x−A) for x≠A. Given ε=9/4>0 and failed δ=36/36>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/4, 9/4)","explanation":"The witness pair is (1/4, 9/4): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0010","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b163c5be55b508942d67bc390d22cba71b7784e1e62a19343390447980e6469d","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093010,"semantic_template":"qf-density-limit-invalid-delta-witness/[10,\"2/1\",\"2/5\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-10)(x−A) for x≠A. Given ε=10/5>0 and failed δ=20/50>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/5, 2/1)","explanation":"The witness pair is (1/5, 2/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0011","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e3cee20d3cb28f70b034416e7da4bedf10e7a8a782192e57ed21aa56995917a","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093011,"semantic_template":"qf-density-limit-invalid-delta-witness/[11,\"11/1\",\"3/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(11)(x−A) for x≠A. Given ε=11/1>0 and failed δ=33/11>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 11/1)","explanation":"The witness pair is (1/1, 11/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0012","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c751ff611823863b8434a6ae4a9b915b99a9623c71e499d25e4c6d0379aa23ee","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093012,"semantic_template":"qf-density-limit-invalid-delta-witness/[12,\"6/1\",\"2/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-12)(x−A) for x≠A. Given ε=12/2>0 and failed δ=48/24>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 6/1)","explanation":"The witness pair is (1/2, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0013","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"199dec0470d3a35a127f475895b7e5b660cb72d8a14ae343b543add2b71fd9b9","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093013,"semantic_template":"qf-density-limit-invalid-delta-witness/[13,\"13/3\",\"2/3\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(13)(x−A) for x≠A. Given ε=13/3>0 and failed δ=26/39>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 13/3)","explanation":"The witness pair is (1/3, 13/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0014","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1d6a2b4a19c51084aa7a9278ba1122260759fe4ae6e12300c1661358f6eafa1","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093014,"semantic_template":"qf-density-limit-invalid-delta-witness/[14,\"7/2\",\"3/4\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-14)(x−A) for x≠A. Given ε=14/4>0 and failed δ=42/56>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/4, 7/2)","explanation":"The witness pair is (1/4, 7/2): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0015","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f43418983bfea9976ed862425b1c3f29bc48b2485438ddcdf4fabcfc6c0ddd61","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093015,"semantic_template":"qf-density-limit-invalid-delta-witness/[15,\"3/1\",\"4/5\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(15)(x−A) for x≠A. Given ε=15/5>0 and failed δ=60/75>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/5, 3/1)","explanation":"The witness pair is (1/5, 3/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0016","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cba8c47b515c67afb779028941df697b607da9709f196723dfed5c03fba59e72","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093016,"semantic_template":"qf-density-limit-invalid-delta-witness/[16,\"16/1\",\"2/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-16)(x−A) for x≠A. Given ε=16/1>0 and failed δ=32/16>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 16/1)","explanation":"The witness pair is (1/1, 16/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0017","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39bf273b0a999a0c4ce4c15c75ce415264cd7570382947fb399d0e1910f401c1","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093017,"semantic_template":"qf-density-limit-invalid-delta-witness/[17,\"17/2\",\"3/2\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(17)(x−A) for x≠A. Given ε=17/2>0 and failed δ=51/34>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 17/2)","explanation":"The witness pair is (1/2, 17/2): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0018","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c39a4324a171e3f80945521a1865fc46924bc99afc4220371b0c18b37f61cbc","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093018,"semantic_template":"qf-density-limit-invalid-delta-witness/[18,\"6/1\",\"4/3\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-18)(x−A) for x≠A. Given ε=18/3>0 and failed δ=72/54>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 6/1)","explanation":"The witness pair is (1/3, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0019","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c866cd01db79ed2de15206c2d362ded2b4e72907001c772e7cc7fe10b1ad20e","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093019,"semantic_template":"qf-density-limit-invalid-delta-witness/[19,\"19/4\",\"1/2\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(19)(x−A) for x≠A. Given ε=19/4>0 and failed δ=38/76>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/4, 19/4)","explanation":"The witness pair is (1/4, 19/4): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0020","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78622e102519f5ad22030f9b85facc36535c5ec9ca84a97ab6e5b2488cb28599","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093020,"semantic_template":"qf-density-limit-invalid-delta-witness/[20,\"4/1\",\"3/5\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-20)(x−A) for x≠A. Given ε=20/5>0 and failed δ=60/100>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/5, 4/1)","explanation":"The witness pair is (1/5, 4/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0021","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f549af9f4b4627a6198e977aff000ef2b26dd7697ddb77157b63d92f0cfb565","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093021,"semantic_template":"qf-density-limit-invalid-delta-witness/[21,\"21/1\",\"4/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(21)(x−A) for x≠A. Given ε=21/1>0 and failed δ=84/21>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 21/1)","explanation":"The witness pair is (1/1, 21/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0022","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e0d6aef2de6df1f0429842cdbacf483204432ca0968cd11a968405a901e7500","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093022,"semantic_template":"qf-density-limit-invalid-delta-witness/[22,\"11/1\",\"1/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-22)(x−A) for x≠A. Given ε=22/2>0 and failed δ=44/44>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 11/1)","explanation":"The witness pair is (1/2, 11/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0023","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31f30c91b9f7d12b66bbe7c6814f2203f03db23748f6e35533e1eccd614b0205","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093023,"semantic_template":"qf-density-limit-invalid-delta-witness/[23,\"23/3\",\"1/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(23)(x−A) for x≠A. Given ε=23/3>0 and failed δ=69/69>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 23/3)","explanation":"The witness pair is (1/3, 23/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0024","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb498b53c04b0dd28a2a867dafa0ef30508cd5c1b310f24108951aca987283b8","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093024,"semantic_template":"qf-density-limit-invalid-delta-witness/[24,\"6/1\",\"1/1\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(-24)(x−A) for x≠A. Given ε=24/4>0 and failed δ=96/96>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/4, 6/1)","explanation":"The witness pair is (1/4, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0025","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c807e4ea6c9619a413de6407460aa97df84773aee621384381b248276618fec","family_id":"qf-density-limit-invalid-delta-witness","seed":202610093025,"semantic_template":"qf-density-limit-invalid-delta-witness/[25,\"5/1\",\"2/5\"]"},"fields":{"title":"Exact Punctured Counterexample to a Failed δ","stem":"On a punctured real domain, f(x)=L+(25)(x−A) for x≠A. Given ε=25/5>0 and failed δ=50/125>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/5, 5/1)","explanation":"The witness pair is (1/5, 5/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required."},"skills":["Exact Punctured Counterexample to a Failed δ"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0026","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61cd7b42954e713c17f65910fb2a0317d92bb0fa371592b789b97698da62990f","family_id":"qf-density-limit-center-versus-limit","seed":202610093026,"semantic_template":"qf-density-limit-center-versus-limit/[-12,\"-12\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-12)x+(-36) away from a=-2, and separately defined center value -12. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-12, -12)","explanation":"The explicit point value and affine approach value give (-12, -12), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0027","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e52ada30a72471dd4419edf1a60ac47a96abfe0c33c1fd5890e630a3886a0bff","family_id":"qf-density-limit-center-versus-limit","seed":202610093027,"semantic_template":"qf-density-limit-center-versus-limit/[-9,\"-11\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-11)x+(-22) away from a=-1, and separately defined center value -9. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-9, -11)","explanation":"The explicit point value and affine approach value give (-9, -11), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0028","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81a1fdca692a4dd562bb1bb18abb6b10e6bbcf5381e4a561bd97a5504f4304ca","family_id":"qf-density-limit-center-versus-limit","seed":202610093028,"semantic_template":"qf-density-limit-center-versus-limit/[-7,\"-10\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-10)x+(-10) away from a=0, and separately defined center value -7. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-7, -10)","explanation":"The explicit point value and affine approach value give (-7, -10), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0029","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"993ae1929472250c936618fce0e2f2aed7138dcb106ca00d34cc8d691134dea5","family_id":"qf-density-limit-center-versus-limit","seed":202610093029,"semantic_template":"qf-density-limit-center-versus-limit/[-9,\"-9\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-9)x+(0) away from a=1, and separately defined center value -9. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-9, -9)","explanation":"The explicit point value and affine approach value give (-9, -9), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0030","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9441b2f6fc10fdc290675812539d592b9da8c663e0aa9e3aca39d6533e80ed3","family_id":"qf-density-limit-center-versus-limit","seed":202610093030,"semantic_template":"qf-density-limit-center-versus-limit/[-3,\"-8\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-8)x+(8) away from a=2, and separately defined center value -3. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-3, -8)","explanation":"The explicit point value and affine approach value give (-3, -8), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0031","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"757e81e296ef356a421ee98e10b9ecd3be1987256b2f3d26abf3055155a68808","family_id":"qf-density-limit-center-versus-limit","seed":202610093031,"semantic_template":"qf-density-limit-center-versus-limit/[-6,\"-7\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-7)x+(-21) away from a=-2, and separately defined center value -6. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-6, -7)","explanation":"The explicit point value and affine approach value give (-6, -7), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0032","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d7b86bfba9b31415a99f1c5b179d3c7df930751d7c8db78fd0b1c887f11f152","family_id":"qf-density-limit-center-versus-limit","seed":202610093032,"semantic_template":"qf-density-limit-center-versus-limit/[-6,\"-6\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-6)x+(-12) away from a=-1, and separately defined center value -6. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-6, -6)","explanation":"The explicit point value and affine approach value give (-6, -6), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0033","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8437b1ef9a067c00c0650434f3c66cd7e6c1be43666e844e23a10761da0064f","family_id":"qf-density-limit-center-versus-limit","seed":202610093033,"semantic_template":"qf-density-limit-center-versus-limit/[-2,\"-5\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-5)x+(-5) away from a=0, and separately defined center value -2. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-2, -5)","explanation":"The explicit point value and affine approach value give (-2, -5), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0034","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e616a3093f26a97232d8bde9cd9503f8ba0c2190696cf327ac980177f5ca527","family_id":"qf-density-limit-center-versus-limit","seed":202610093034,"semantic_template":"qf-density-limit-center-versus-limit/[0,\"-4\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-4)x+(0) away from a=1, and separately defined center value 0. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(0, -4)","explanation":"The explicit point value and affine approach value give (0, -4), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0035","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02a911277883e6c29558adfda1b85ba4e41258e533d260010e627eab434b29ce","family_id":"qf-density-limit-center-versus-limit","seed":202610093035,"semantic_template":"qf-density-limit-center-versus-limit/[-3,\"-3\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-3)x+(3) away from a=2, and separately defined center value -3. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-3, -3)","explanation":"The explicit point value and affine approach value give (-3, -3), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0036","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cfd78e99419ee93c49174b073d2d59b803a234241c629e14bdfe8b6fadd32aa","family_id":"qf-density-limit-center-versus-limit","seed":202610093036,"semantic_template":"qf-density-limit-center-versus-limit/[-1,\"-2\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-2)x+(-6) away from a=-2, and separately defined center value -1. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(-1, -2)","explanation":"The explicit point value and affine approach value give (-1, -2), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0037","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b239834dc94d2459186eb6504666e3abbb62387418e22013cea663524262f8","family_id":"qf-density-limit-center-versus-limit","seed":202610093037,"semantic_template":"qf-density-limit-center-versus-limit/[1,\"-1\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (-1)x+(-2) away from a=-1, and separately defined center value 1. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(1, -1)","explanation":"The explicit point value and affine approach value give (1, -1), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0038","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d939817887a8c1b9ceee18b358170b8b43e6d2a19be46957133e499d4ca07250","family_id":"qf-density-limit-center-versus-limit","seed":202610093038,"semantic_template":"qf-density-limit-center-versus-limit/[0,\"0\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (0)x+(0) away from a=0, and separately defined center value 0. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(0, 0)","explanation":"The explicit point value and affine approach value give (0, 0), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0039","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d5405caafa9241d4d43faf189965cf16e88ff6d425b2dee368b516bc730ddcb","family_id":"qf-density-limit-center-versus-limit","seed":202610093039,"semantic_template":"qf-density-limit-center-versus-limit/[5,\"1\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (1)x+(0) away from a=1, and separately defined center value 5. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(5, 1)","explanation":"The explicit point value and affine approach value give (5, 1), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0040","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a79cfc6506affe14aa5b98f35adca3ce7358ebb222d531d745340425b6c142e7","family_id":"qf-density-limit-center-versus-limit","seed":202610093040,"semantic_template":"qf-density-limit-center-versus-limit/[7,\"2\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (2)x+(-2) away from a=2, and separately defined center value 7. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(7, 2)","explanation":"The explicit point value and affine approach value give (7, 2), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0041","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c70670f77f2b3aee05f62b6abc62c81e5e06184f346125303d1e8119ed2ce54","family_id":"qf-density-limit-center-versus-limit","seed":202610093041,"semantic_template":"qf-density-limit-center-versus-limit/[3,\"3\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (3)x+(9) away from a=-2, and separately defined center value 3. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(3, 3)","explanation":"The explicit point value and affine approach value give (3, 3), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0042","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b05597a39b4cd438776562942ce0e08c4ac4df9b51d4abbdc79d354abf61fd87","family_id":"qf-density-limit-center-versus-limit","seed":202610093042,"semantic_template":"qf-density-limit-center-versus-limit/[6,\"4\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (4)x+(8) away from a=-1, and separately defined center value 6. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(6, 4)","explanation":"The explicit point value and affine approach value give (6, 4), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0043","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5123e8bdbd78fd8fe9739759ba8486c7e859bcff1e2f3caeb7ea2195ce2744d","family_id":"qf-density-limit-center-versus-limit","seed":202610093043,"semantic_template":"qf-density-limit-center-versus-limit/[8,\"5\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (5)x+(5) away from a=0, and separately defined center value 8. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(8, 5)","explanation":"The explicit point value and affine approach value give (8, 5), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0044","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3addf76e2904934bf99cc2cc0d645a9d1d7b02f5872be50cd8270b6abc4c87f6","family_id":"qf-density-limit-center-versus-limit","seed":202610093044,"semantic_template":"qf-density-limit-center-versus-limit/[6,\"6\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (6)x+(0) away from a=1, and separately defined center value 6. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(6, 6)","explanation":"The explicit point value and affine approach value give (6, 6), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0045","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96a3ef2ace9ced337347bdab786c7e585c8e4d52ce696b3cbffa110b0cd73857","family_id":"qf-density-limit-center-versus-limit","seed":202610093045,"semantic_template":"qf-density-limit-center-versus-limit/[12,\"7\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (7)x+(-7) away from a=2, and separately defined center value 12. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(12, 7)","explanation":"The explicit point value and affine approach value give (12, 7), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0046","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09884597a92abd3936941f48401b9a7d064a2ed14dfdcb11a694cac15b511282","family_id":"qf-density-limit-center-versus-limit","seed":202610093046,"semantic_template":"qf-density-limit-center-versus-limit/[9,\"8\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (8)x+(24) away from a=-2, and separately defined center value 9. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(9, 8)","explanation":"The explicit point value and affine approach value give (9, 8), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0047","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2591f7bfd62c662271f89910fc1bc3677036867e2e380e03059da14dd799e8ee","family_id":"qf-density-limit-center-versus-limit","seed":202610093047,"semantic_template":"qf-density-limit-center-versus-limit/[9,\"9\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (9)x+(18) away from a=-1, and separately defined center value 9. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(9, 9)","explanation":"The explicit point value and affine approach value give (9, 9), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0048","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07a1287cdda60d7108063f95173610c92d1666039da3e05ee3195ab6e2fb38cc","family_id":"qf-density-limit-center-versus-limit","seed":202610093048,"semantic_template":"qf-density-limit-center-versus-limit/[13,\"10\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (10)x+(10) away from a=0, and separately defined center value 13. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(13, 10)","explanation":"The explicit point value and affine approach value give (13, 10), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0049","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54617023899034fd4c0d4abf5c418bc75070138dfebfb1bf020e8fd44b4c2e19","family_id":"qf-density-limit-center-versus-limit","seed":202610093049,"semantic_template":"qf-density-limit-center-versus-limit/[15,\"11\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (11)x+(0) away from a=1, and separately defined center value 15. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(15, 11)","explanation":"The explicit point value and affine approach value give (15, 11), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0050","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d6a2ea7ca8afbdc3c1308fb878f104ab1c02fee6ab5d2324f2ebe096bb00824","family_id":"qf-density-limit-center-versus-limit","seed":202610093050,"semantic_template":"qf-density-limit-center-versus-limit/[12,\"12\"]"},"fields":{"title":"Ordered Center-Value and Punctured-Limit Pair","stem":"The full-real function has affine expression (12)x+(-12) away from a=2, and separately defined center value 12. Give (center value,punctured two-sided limit).","answer":{"kind":"expression","value":"(12, 12)","explanation":"The explicit point value and affine approach value give (12, 12), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity."},"skills":["Ordered Center-Value and Punctured-Limit Pair"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0051","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09de08c4cae768ce6cc41b36124a8e4789483948efba3a3f8eff22980c6a0c70","family_id":"qf-density-limit-one-sided-pair","seed":202610093051,"semantic_template":"qf-density-limit-one-sided-pair/[\"-12\",\"-12\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (1)x+(-10), and for x>a is (-1)x+(-14). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-12, -12)","explanation":"Evaluate the two affine approach values to obtain (-12, -12). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0052","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b69e3da673cb291e61c8f42f8cfc9c4f2888d0cc66aa9ca0184b2058f2b5d896","family_id":"qf-density-limit-one-sided-pair","seed":202610093052,"semantic_template":"qf-density-limit-one-sided-pair/[\"-11\",\"12\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-1}, the branch for x<a is (2)x+(-9), and for x>a is (-2)x+(10). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-11, 12)","explanation":"Evaluate the two affine approach values to obtain (-11, 12). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0053","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef1144d7f81ef904a10838bbb51323a4a88890ca844ecf01fc443730be29fdc","family_id":"qf-density-limit-one-sided-pair","seed":202610093053,"semantic_template":"qf-density-limit-one-sided-pair/[\"-10\",\"11\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{0}, the branch for x<a is (3)x+(-10), and for x>a is (-1)x+(11). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-10, 11)","explanation":"Evaluate the two affine approach values to obtain (-10, 11). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0054","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b015040b98a00e31459ac784038f49519721d879b9db77cfbe0c3d3c8c2bc80","family_id":"qf-density-limit-one-sided-pair","seed":202610093054,"semantic_template":"qf-density-limit-one-sided-pair/[\"-9\",\"-9\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{1}, the branch for x<a is (1)x+(-10), and for x>a is (-2)x+(-7). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-9, -9)","explanation":"Evaluate the two affine approach values to obtain (-9, -9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0055","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4629c12866d166b262c22e3c1d1e9fe2f54ade5e12f2f0eff958e49619ffc543","family_id":"qf-density-limit-one-sided-pair","seed":202610093055,"semantic_template":"qf-density-limit-one-sided-pair/[\"-8\",\"9\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (2)x+(-4), and for x>a is (-1)x+(7). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-8, 9)","explanation":"Evaluate the two affine approach values to obtain (-8, 9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0056","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c004f93b6240e0aabb0aa22bc86c2d3595a305bc3ca19617e185c28a1c3f62c1","family_id":"qf-density-limit-one-sided-pair","seed":202610093056,"semantic_template":"qf-density-limit-one-sided-pair/[\"-7\",\"8\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-1}, the branch for x<a is (3)x+(-4), and for x>a is (-2)x+(6). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-7, 8)","explanation":"Evaluate the two affine approach values to obtain (-7, 8). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0057","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9feda3a872c65819a3cdcb6ba58043e9ca03e01985a192d28e9bede465102e53","family_id":"qf-density-limit-one-sided-pair","seed":202610093057,"semantic_template":"qf-density-limit-one-sided-pair/[\"-6\",\"-6\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{0}, the branch for x<a is (1)x+(-6), and for x>a is (-1)x+(-6). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-6, -6)","explanation":"Evaluate the two affine approach values to obtain (-6, -6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0058","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8108a5be21158a251dd12be223cdcb4556e5eebe41455d142eb42373a7c2f66f","family_id":"qf-density-limit-one-sided-pair","seed":202610093058,"semantic_template":"qf-density-limit-one-sided-pair/[\"-5\",\"6\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{1}, the branch for x<a is (2)x+(-7), and for x>a is (-2)x+(8). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-5, 6)","explanation":"Evaluate the two affine approach values to obtain (-5, 6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0059","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8f6476f91d366eabeca2a6cf88b25018ae94e666f21d0efe7c4ee5b06c60e46","family_id":"qf-density-limit-one-sided-pair","seed":202610093059,"semantic_template":"qf-density-limit-one-sided-pair/[\"-4\",\"5\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (3)x+(2), and for x>a is (-1)x+(3). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-4, 5)","explanation":"Evaluate the two affine approach values to obtain (-4, 5). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0060","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a863523496cf703c16c7d06151ffa4f85fd49b7c82e7786724353fea7061f1a","family_id":"qf-density-limit-one-sided-pair","seed":202610093060,"semantic_template":"qf-density-limit-one-sided-pair/[\"-3\",\"-3\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-1}, the branch for x<a is (1)x+(-2), and for x>a is (-2)x+(-5). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-3, -3)","explanation":"Evaluate the two affine approach values to obtain (-3, -3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0061","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8f9da7b38f021e6b43166f0573ab96c91f9d3f3c127035db9949c6811e71555","family_id":"qf-density-limit-one-sided-pair","seed":202610093061,"semantic_template":"qf-density-limit-one-sided-pair/[\"-2\",\"3\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{0}, the branch for x<a is (2)x+(-2), and for x>a is (-1)x+(3). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-2, 3)","explanation":"Evaluate the two affine approach values to obtain (-2, 3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0062","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c290c34b2ac1ce61f88910a238989aca497351629b3038a0b3a755b46ed336","family_id":"qf-density-limit-one-sided-pair","seed":202610093062,"semantic_template":"qf-density-limit-one-sided-pair/[\"-1\",\"2\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{1}, the branch for x<a is (3)x+(-4), and for x>a is (-2)x+(4). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(-1, 2)","explanation":"Evaluate the two affine approach values to obtain (-1, 2). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0063","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"256a830c013772d3745066e6d833a08397544fee4a9556ba5aa3e8232c1d3738","family_id":"qf-density-limit-one-sided-pair","seed":202610093063,"semantic_template":"qf-density-limit-one-sided-pair/[\"0\",\"0\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (1)x+(2), and for x>a is (-1)x+(-2). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Evaluate the two affine approach values to obtain (0, 0). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0064","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f59d3a84247db91c15a977b30b388b463ecb99bd82259e0141ca9c8270e0991","family_id":"qf-density-limit-one-sided-pair","seed":202610093064,"semantic_template":"qf-density-limit-one-sided-pair/[\"1\",\"0\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-1}, the branch for x<a is (2)x+(3), and for x>a is (-2)x+(-2). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(1, 0)","explanation":"Evaluate the two affine approach values to obtain (1, 0). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0065","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0308232c2e0dadb2e5b408f421d64c49fb192cbe2ca766da772c17dc60688e5b","family_id":"qf-density-limit-one-sided-pair","seed":202610093065,"semantic_template":"qf-density-limit-one-sided-pair/[\"2\",\"-1\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{0}, the branch for x<a is (3)x+(2), and for x>a is (-1)x+(-1). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(2, -1)","explanation":"Evaluate the two affine approach values to obtain (2, -1). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0066","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e53869e29418a5f5161a17c4c874dab7ba9dfc2dbc88f6fd213ab499459c82f","family_id":"qf-density-limit-one-sided-pair","seed":202610093066,"semantic_template":"qf-density-limit-one-sided-pair/[\"3\",\"3\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{1}, the branch for x<a is (1)x+(2), and for x>a is (-2)x+(5). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(3, 3)","explanation":"Evaluate the two affine approach values to obtain (3, 3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0067","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01f5ae33ecbfe282ded127fb46d8fbf5848b461a9309f5e9f0d2c2f845fafdb1","family_id":"qf-density-limit-one-sided-pair","seed":202610093067,"semantic_template":"qf-density-limit-one-sided-pair/[\"4\",\"-3\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (2)x+(8), and for x>a is (-1)x+(-5). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(4, -3)","explanation":"Evaluate the two affine approach values to obtain (4, -3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0068","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53cdd68a1828561689e4aa75476d6a5462d8586c05d873affad24f6dcd2c1e9b","family_id":"qf-density-limit-one-sided-pair","seed":202610093068,"semantic_template":"qf-density-limit-one-sided-pair/[\"5\",\"-4\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-1}, the branch for x<a is (3)x+(8), and for x>a is (-2)x+(-6). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(5, -4)","explanation":"Evaluate the two affine approach values to obtain (5, -4). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0069","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78b51d2e5088230d3861c62ed93dc8394e5b607c8db614a7183a44eaf92e0353","family_id":"qf-density-limit-one-sided-pair","seed":202610093069,"semantic_template":"qf-density-limit-one-sided-pair/[\"6\",\"6\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{0}, the branch for x<a is (1)x+(6), and for x>a is (-1)x+(6). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(6, 6)","explanation":"Evaluate the two affine approach values to obtain (6, 6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0070","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb64d0485585f9a96b69fcd6ee2730cbea67038297c8651d7d6da2f9b2d74a7e","family_id":"qf-density-limit-one-sided-pair","seed":202610093070,"semantic_template":"qf-density-limit-one-sided-pair/[\"7\",\"-6\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{1}, the branch for x<a is (2)x+(5), and for x>a is (-2)x+(-4). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(7, -6)","explanation":"Evaluate the two affine approach values to obtain (7, -6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0071","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1dbc316f8d64ad9ff95588d272224b082322e3718348383ba42a2728c33408e","family_id":"qf-density-limit-one-sided-pair","seed":202610093071,"semantic_template":"qf-density-limit-one-sided-pair/[\"8\",\"-7\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (3)x+(14), and for x>a is (-1)x+(-9). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(8, -7)","explanation":"Evaluate the two affine approach values to obtain (8, -7). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0072","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a44b5b7340eafa80e3be1c36183828e964dc06ec717c8db08febc209b2e1971f","family_id":"qf-density-limit-one-sided-pair","seed":202610093072,"semantic_template":"qf-density-limit-one-sided-pair/[\"9\",\"9\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-1}, the branch for x<a is (1)x+(10), and for x>a is (-2)x+(7). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(9, 9)","explanation":"Evaluate the two affine approach values to obtain (9, 9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0073","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b925493ac225f1a81ecc9ba507d75ac3688cef844aa1694ce46aba28d1e503","family_id":"qf-density-limit-one-sided-pair","seed":202610093073,"semantic_template":"qf-density-limit-one-sided-pair/[\"10\",\"-9\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{0}, the branch for x<a is (2)x+(10), and for x>a is (-1)x+(-9). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(10, -9)","explanation":"Evaluate the two affine approach values to obtain (10, -9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0074","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a3651508676515a990cc59f8375dc58709f8a0cde9ce50629e0db6b51fa7e46","family_id":"qf-density-limit-one-sided-pair","seed":202610093074,"semantic_template":"qf-density-limit-one-sided-pair/[\"11\",\"-10\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{1}, the branch for x<a is (3)x+(8), and for x>a is (-2)x+(-8). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(11, -10)","explanation":"Evaluate the two affine approach values to obtain (11, -10). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0075","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdfde7ef6cc332e5d57dc39cef55639d514797c5d75a002f97df706cda14077a","family_id":"qf-density-limit-one-sided-pair","seed":202610093075,"semantic_template":"qf-density-limit-one-sided-pair/[\"12\",\"12\"]"},"fields":{"title":"Ordered Left and Right Punctured Limits","stem":"On R∖{-2}, the branch for x<a is (1)x+(14), and for x>a is (-1)x+(10). Give (left limit,right limit), keeping the order.","answer":{"kind":"expression","value":"(12, 12)","explanation":"Evaluate the two affine approach values to obtain (12, 12). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful."},"skills":["Ordered Left and Right Punctured Limits"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0076","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98c080e7713f8f7579e18a0c530e6b15046fc087fc627dcff9000d6d0d709c13","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093076,"semantic_template":"qf-density-limit-punctured-quotient-value/[-12,-11]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-12)²)/(x−(-12)) has punctured real domain x≠a. Evaluate it at the permitted x=-11. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-23/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -23/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0077","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc2e28a09a00f47ccb72a2f5329a8e7a5c5128514b7ae8673f8bb3452c7916c0","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093077,"semantic_template":"qf-density-limit-punctured-quotient-value/[-11,-13]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-11)²)/(x−(-11)) has punctured real domain x≠a. Evaluate it at the permitted x=-13. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-24/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -24/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0078","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c107923030327a205dbf76df2af30f45ad0f9d7b0fce8f8eabb7999022ad3238","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093078,"semantic_template":"qf-density-limit-punctured-quotient-value/[-10,-7]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-10)²)/(x−(-10)) has punctured real domain x≠a. Evaluate it at the permitted x=-7. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-17/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -17/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0079","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6888b13d228ae59d2672a03b179d39b66ca839bd05cd6eeb2a11b960ab316411","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093079,"semantic_template":"qf-density-limit-punctured-quotient-value/[-9,-13]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-9)²)/(x−(-9)) has punctured real domain x≠a. Evaluate it at the permitted x=-13. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-22/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -22/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0080","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"718e05c5802c1fac47b14f61e533155813fc580ef5636c1451efbfb5c750c3d0","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093080,"semantic_template":"qf-density-limit-punctured-quotient-value/[-8,-7]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-8)²)/(x−(-8)) has punctured real domain x≠a. Evaluate it at the permitted x=-7. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-15/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -15/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0081","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"366d120316d73da3b13b13331e7c5c158050d03a5fe9a2cc3783d5705b921764","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093081,"semantic_template":"qf-density-limit-punctured-quotient-value/[-7,-9]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-7)²)/(x−(-7)) has punctured real domain x≠a. Evaluate it at the permitted x=-9. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-16/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -16/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0082","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4debb3dd4ba461988d9504f534729c2268ed08e4f8a63d86201dab08f7b89829","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093082,"semantic_template":"qf-density-limit-punctured-quotient-value/[-6,-3]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-6)²)/(x−(-6)) has punctured real domain x≠a. Evaluate it at the permitted x=-3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-9/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -9/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0083","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25aad6f7801eb53cb0aa287b6bc34699fa7983430598f9b422e7382c80122ecb","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093083,"semantic_template":"qf-density-limit-punctured-quotient-value/[-5,-9]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-5)²)/(x−(-5)) has punctured real domain x≠a. Evaluate it at the permitted x=-9. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-14/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -14/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0084","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dda6bd80bc8e8afe754d735303ffdcdac15c5b5fe7018031100e4575ff6ca98","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093084,"semantic_template":"qf-density-limit-punctured-quotient-value/[-4,-3]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-4)²)/(x−(-4)) has punctured real domain x≠a. Evaluate it at the permitted x=-3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-7/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -7/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0085","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"862b75dc5ae3e6a780c14c6772c73718fa15e65cc784c1d8113fabd510fc4635","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093085,"semantic_template":"qf-density-limit-punctured-quotient-value/[-3,-5]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-3)²)/(x−(-3)) has punctured real domain x≠a. Evaluate it at the permitted x=-5. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -8/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0086","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"577f2e0b32b4f281ffecf1dcc04af9d17b147b31873f2ee3be2bf4db4d683133","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093086,"semantic_template":"qf-density-limit-punctured-quotient-value/[-2,1]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-2)²)/(x−(-2)) has punctured real domain x≠a. Evaluate it at the permitted x=1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-1/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -1/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0087","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"851b4e7daa214a9298bc8792b04c38e6bc8349bb90830bea33b25e9e2b36c4d8","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093087,"semantic_template":"qf-density-limit-punctured-quotient-value/[-1,-5]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(-1)²)/(x−(-1)) has punctured real domain x≠a. Evaluate it at the permitted x=-5. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-6/1","explanation":"The denominator is nonzero here, so the exact original quotient value is -6/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0088","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3de77dc4bb3b6b698fdf5b5c338f46ae6ab45164b32ea6b3ccdbde0cc45d7f18","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093088,"semantic_template":"qf-density-limit-punctured-quotient-value/[0,1]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(0)²)/(x−(0)) has punctured real domain x≠a. Evaluate it at the permitted x=1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 1/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0089","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b0404dc7bac423cacc51064e2e57505faaa0855784f846a5af1f5cb15c634b1","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093089,"semantic_template":"qf-density-limit-punctured-quotient-value/[1,-1]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(1)²)/(x−(1)) has punctured real domain x≠a. Evaluate it at the permitted x=-1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 0/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0090","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40b7d9e1e1d57a876bdb77859d16f94a3d500e7c6032226dd6884585b9d029c0","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093090,"semantic_template":"qf-density-limit-punctured-quotient-value/[2,5]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(2)²)/(x−(2)) has punctured real domain x≠a. Evaluate it at the permitted x=5. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"7/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 7/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0091","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"762e1a29d2804b8cfcab32b303e25bb877b9bfdc3beaf8598f9e86ed249ec0f2","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093091,"semantic_template":"qf-density-limit-punctured-quotient-value/[3,-1]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(3)²)/(x−(3)) has punctured real domain x≠a. Evaluate it at the permitted x=-1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 2/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0092","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b4b0fc0d7da8ade6e54231c055d0762dd4d7508ac3a5deafe1e41196820b11e","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093092,"semantic_template":"qf-density-limit-punctured-quotient-value/[4,5]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(4)²)/(x−(4)) has punctured real domain x≠a. Evaluate it at the permitted x=5. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"9/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 9/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0093","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"075c503f2d8831139429b86d2b25108a4b66342a615c8a5eaf12058a51fe4737","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093093,"semantic_template":"qf-density-limit-punctured-quotient-value/[5,3]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(5)²)/(x−(5)) has punctured real domain x≠a. Evaluate it at the permitted x=3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"8/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 8/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0094","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b9633b0e89187a7612803423eb90e3a35c7ece3cf54b648c7c5e3f51e34847","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093094,"semantic_template":"qf-density-limit-punctured-quotient-value/[6,9]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(6)²)/(x−(6)) has punctured real domain x≠a. Evaluate it at the permitted x=9. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"15/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 15/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0095","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c277a1153c5065ed995d98fb06ddc35e5de6327407ae2075cd84243cfd51aab3","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093095,"semantic_template":"qf-density-limit-punctured-quotient-value/[7,3]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(7)²)/(x−(7)) has punctured real domain x≠a. Evaluate it at the permitted x=3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"10/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 10/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0096","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d23854fb90e73a19b9a4e277355b48ff1426592dae1fe1c2685f1474804d6d3b","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093096,"semantic_template":"qf-density-limit-punctured-quotient-value/[8,9]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(8)²)/(x−(8)) has punctured real domain x≠a. Evaluate it at the permitted x=9. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"17/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 17/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0097","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ba88f07e3a22e0e53fe2fd19a02230a6138aba489591b8f73bbf8104513e9da","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093097,"semantic_template":"qf-density-limit-punctured-quotient-value/[9,7]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(9)²)/(x−(9)) has punctured real domain x≠a. Evaluate it at the permitted x=7. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"16/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 16/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0098","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d16c5aa8d8938ddfd53e59c246514e8285580b70103fbaf96fd22ccf0d250e3a","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093098,"semantic_template":"qf-density-limit-punctured-quotient-value/[10,13]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(10)²)/(x−(10)) has punctured real domain x≠a. Evaluate it at the permitted x=13. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"23/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 23/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0099","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e80952541f8b08948a8478556f45ccd7e9e439337eb6d13111a195c963cd9d7","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093099,"semantic_template":"qf-density-limit-punctured-quotient-value/[11,7]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(11)²)/(x−(11)) has punctured real domain x≠a. Evaluate it at the permitted x=7. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"18/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 18/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r03-function-limit-0100","batch_id":"batch-20261007-advance-20261009-r03-function-limit-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d04cc2d81f0a0df1454357ab61914fa508d7f3d9976f10535497c016a75049e","family_id":"qf-density-limit-punctured-quotient-value","seed":202610093100,"semantic_template":"qf-density-limit-punctured-quotient-value/[12,13]"},"fields":{"title":"Original Quotient Value at a Permitted Punctured Point","stem":"The original quotient (x²−(12)²)/(x−(12)) has punctured real domain x≠a. Evaluate it at the permitted x=13. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"25/1","explanation":"The denominator is nonzero here, so the exact original quotient value is 25/1. This does not fill the excluded center or assign a division-by-zero default."},"skills":["Original Quotient Value at a Permitted Punctured Point"],"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."}}