# Question 23610: Both Sides of the Interior Singularity Need Independent Limits [Return to catalogue page 119](https://ocme.evemisslab.com/en/questions/page/119/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r04-improper-integral-foundations/#q-20261014-advance-20261112-r04-improper-integral-0035) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0046-23610 · q-20261014-advance-20261112-r04-improper-integral-0035 ## Original question and givens Use c=-3, interval [c−1,c+1], left integrand=10*(c−x)^(−1/2) and right integrand=11*(x−c)^(−2/3). Both exponents are reduced rationals in [0,1). Give (interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate). - G001 — Use c=-3, interval [c−1,c+1], left integrand=10*(c−x)^(−1/2) and right integrand=11*(x−c)^(−2/3). Both exponents are reduced rationals in [0,1). Give (interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-improper-integral](https://ocme.evemisslab.com/en/mko/mko-improper-integral/) ## Step-by-step derivation S001 · impliesBefore source operation=improper-interior-singular-split;original snapshot and mathematics retained After explicit ordered source output fields=[interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate] Conditions: none Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore Improper LEFT reduced exponent numerator gap := sub(2, 1) After Improper LEFT reduced exponent numerator gap = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Improper LEFT reciprocal of one minus exponent := div(2, 1) After Improper LEFT reciprocal of one minus exponent = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Improper LEFT scale times singular-side integral := mul(10, 2) After Improper LEFT scale times singular-side integral = 20 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore ORIGINAL LEFT side at c=-3;scale=10;REDUCED exponent=1/2 in[0,1) After put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=20 Conditions: none Use positive distance from the singularity on each side and take its own delta->0 limit.Exponent0 is legal with reduced0/1.Signed/zero scale is allowed;left-right cancellation replaces neither one-sided limit. Step identifiers and dependenciesRule: improper_EACH_interior_singular_side_independent_limit · Uses: G001, S004 S006 · calculationBefore Improper RIGHT reduced exponent numerator gap := sub(3, 2) After Improper RIGHT reduced exponent numerator gap = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Improper RIGHT reciprocal of one minus exponent := div(3, 1) After Improper RIGHT reciprocal of one minus exponent = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Improper RIGHT scale times singular-side integral := mul(11, 3) After Improper RIGHT scale times singular-side integral = 33 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore ORIGINAL RIGHT side at c=-3;scale=11;REDUCED exponent=2/3 in[0,1) After put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=33 Conditions: none Use positive distance from the singularity on each side and take its own delta->0 limit.Exponent0 is legal with reduced0/1.Signed/zero scale is allowed;left-right cancellation replaces neither one-sided limit. Step identifiers and dependenciesRule: improper_EACH_interior_singular_side_independent_limit · Uses: G001, S008 S010 · calculationBefore Improper independent singular-side limits sum := sum(20, 33) After Improper independent singular-side limits sum = 53 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Recompute Improper independent singular-side limits sum with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute Improper RIGHT scale times singular-side integral with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Improper RIGHT reciprocal of one minus exponent with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Improper RIGHT reduced exponent numerator gap with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute Improper LEFT scale times singular-side integral with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Improper LEFT reciprocal of one minus exponent with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Improper LEFT reduced exponent numerator gap with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-4,-2], -3, left:10*(c-x)^(-1/2), right:11*(x-c)^(-2/3), 20, 33, Yes, Yes, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 1ded1a20687219063d8c5b8e263e4109c5f100d5e902e1c56d5842bdb5f5f111 - candidate record SHA-256: 260125bdd6d0ff59fb471298acfeaf52bdcd0b83028f588d5207285eb4c990d2 - semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1 - question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r04-improper-integral-0035.json)