# Question 3910: Ordered Center-Value and Punctured-Limit Pair

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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.

R0006-03910 · q-20261007-advance-20261009-r03-function-limit-0035



## Original question and givens

The full-real function has affine expression (-3)x+(3) away from a=2, and separately defined center value -3. Give (center value,punctured two-sided limit).


- G001 — The full-real function has affine expression (-3)x+(3) away from a=2, and separately defined center value -3. Give (center value,punctured two-sided limit).





## 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

punctured branch slope times center := mul(-3, 2)

After

punctured branch slope times center = -6

Conditions: none

Multiply the branch slope by the center.

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





S003 · calculationBefore

punctured branch limit value := sum(-6, 3)

After

punctured branch limit value = -3

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 · impliesBefore

given f(a)=-3;punctured affine limit=-3

After

ordered pair=(-3,-3)

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.

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









## Reverse checks


- 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.





## Result

(-3, -3)

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: 02a911277883e6c29558adfda1b85ba4e41258e533d260010e627eab434b29ce
- candidate record SHA-256: ce7ae221d2258e5fe6ebe1325e8af8951c9eb6877f19b29139bba3f523f455b2
- 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-0035.json)