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R0006-03966 · q-20261007-advance-20261009-r03-function-limit-0066
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.
G001 — 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.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
left branch slope times center := mul(1, 1)
After
left branch slope times center = 1
Conditions: none
Multiply the branch slope by the center.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
left branch limit value := sum(1, 2)
After
left branch limit value = 3
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
right branch slope times center := mul(-2, 1)
After
right branch slope times center = -2
Conditions: none
Multiply the branch slope by the center.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
right branch limit value := sum(-2, 5)
After
right branch limit value = 3
Conditions: none
Add that branch's constant;the affine difference m(x−a) admits ε−δ control.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left limit=3;right limit=3
After
ordered one-sided pair=(3,3)
Conditions: none
Control each affine side with δ=ε/(1+|m|).Equal side limits give a common two-sided limit;unequal limits do not.Neither case fills a center value.
Rule: one_sided_affine_limit_order · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute right branch limit value with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute right branch slope times center with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute left branch limit value with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute left 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.(3, 3)
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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