# Question 3985: Original Quotient Value at a Permitted Punctured Point [Return to catalogue page 20](https://ocme.evemisslab.com/en/questions/page/20/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r03-function-limit-foundations/#q-20261007-advance-20261009-r03-function-limit-0085) 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. R0006-03985 · q-20261007-advance-20261009-r03-function-limit-0085 ## Original question and givens 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. - G001 — 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. ## 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-function-limit](https://ocme.evemisslab.com/en/mko/mko-function-limit/) ## Step-by-step derivation S001 · impliesBefore declared real model with an excluded center for limit data After point value,legal off-center evaluation and limit are distinct Conditions: none A punctured limit does not read the center value,and evaluation at one legal point is not a limit.These declared affine or factorizable models do not represent arbitrary functions. Step identifiers and dependenciesRule: punctured_limit_scope · Uses: G001 S002 · calculationBefore x squared := square(-5) After x squared = 25 Conditions: none Square the entire specified legal input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore center squared := square(-3) After center squared = 9 Conditions: none Square the entire center,including any negative sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore difference of squares := sub(25, 9) After difference of squares = 16 Conditions: none The numerator is x²−a². Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore nonzero punctured denominator := sub(-5, -3) After nonzero punctured denominator = -2 Conditions: none Check x−a≠0 before division. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore legal quotient value := div(16, -2) After legal quotient value = -8 Conditions: none Divide exactly only at the legal point of the original quotient;do not fill its center value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore x=-5≠a=-3 After (x²−a²)/(x−a)=x+a=-8 Conditions: none Factorization audits the result;the cancellation identity retains x≠a and does not define the original quotient at a. Step identifiers and dependenciesRule: legal_quotient_factor_audit · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute legal quotient value with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute nonzero punctured denominator with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute difference of squares with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute center squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute x squared 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 -8/1 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: 862b75dc5ae3e6a780c14c6772c73718fa15e65cc784c1d8113fabd510fc4635 - candidate record SHA-256: 67f2cf27bb302c739f08c767e59496c19b1302d53933b641fba76ba41a9a943a - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r03-function-limit-0085.json)