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R0006-03912 · q-20261007-advance-20261009-r03-function-limit-0037
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).
G001 — 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).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
punctured branch slope times center := mul(-1, -1)
After
punctured branch slope times center = 1
Conditions: none
Multiply the branch slope by the center.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
punctured branch limit value := sum(1, -2)
After
punctured branch limit value = -1
Conditions: none
Add that branch's constant;the affine difference m(x−a) admits ε−δ control.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
given f(a)=1;punctured affine limit=-1
After
ordered pair=(1,-1)
Conditions: none
The center value is given separately.Off-center f(x)−L=m(x−a),so δ=ε/(1+|m|) proves the limit without changing the point value or assuming continuity.
Rule: point_value_independent_of_limit · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute punctured branch limit value with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute punctured branch slope times center 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.(1, -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.
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