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R0006-03981 · q-20261007-advance-20261009-r03-function-limit-0081
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.
G001 — 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.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.
Before
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.
Rule: punctured_limit_scope · Uses: G001
Before
x squared := square(-9)
After
x squared = 81
Conditions: none
Square the entire specified legal input.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
center squared := square(-7)
After
center squared = 49
Conditions: none
Square the entire center,including any negative sign.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
difference of squares := sub(81, 49)
After
difference of squares = 32
Conditions: none
The numerator is x²−a².
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
nonzero punctured denominator := sub(-9, -7)
After
nonzero punctured denominator = -2
Conditions: none
Check x−a≠0 before division.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
legal quotient value := div(32, -2)
After
legal quotient value = -16
Conditions: none
Divide exactly only at the legal point of the original quotient;do not fill its center value.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
x=-9≠a=-7
After
(x²−a²)/(x−a)=x+a=-16
Conditions: none
Factorization audits the result;the cancellation identity retains x≠a and does not define the original quotient at a.
Rule: legal_quotient_factor_audit · Uses: G001, S006
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.-16/1
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 366d120316d73da3b13b13331e7c5c158050d03a5fe9a2cc3783d5705b921764candidate record SHA-256: 543080df4a4dfaaf51fe3b99e35a7b0fce889275c3e0b4f1c3298944bba9586csemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED