# Question 3952: Ordered Left and Right Punctured Limits

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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.

R0006-03952 · q-20261007-advance-20261009-r03-function-limit-0052



## Original question and givens

On R∖{-1}, the branch for x<a is (2)x+(-9), and for x>a is (-2)x+(10). Give (left limit,right limit), keeping the order.


- G001 — On R∖{-1}, the branch for x<a is (2)x+(-9), and for x>a is (-2)x+(10). Give (left limit,right limit), keeping the order.





## 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

left branch slope times center := mul(2, -1)

After

left branch slope times center = -2

Conditions: none

Multiply the branch slope by the center.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

left branch limit value := sum(-2, -9)

After

left branch limit value = -11

Conditions: none

Add that branch's constant;the affine difference m(x−a) admits ε−δ control.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

right branch slope times center := mul(-2, -1)

After

right branch slope times center = 2

Conditions: none

Multiply the branch slope by the center.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

right branch limit value := sum(2, 10)

After

right branch limit value = 12

Conditions: none

Add that branch's constant;the affine difference m(x−a) admits ε−δ control.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

left limit=-11;right limit=12

After

ordered one-sided pair=(-11,12)

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.

Step identifiers and dependenciesRule: one_sided_affine_limit_order · Uses: G001, S005









## Reverse checks


- 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.





## Result

(-11, 12)

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: b69e3da673cb291e61c8f42f8cfc9c4f2888d0cc66aa9ca0184b2058f2b5d896
- candidate record SHA-256: 70ca865801c06e904eae1ae033357de3eb28c5797ef3c1bed98ca5abf818d598
- 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-0052.json)