# Question 32258: Affine Limit on a Punctured Real Domain

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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.

R0063-32258 · q-20261007-advance-20261008-r02-function-limit-0008



## Original question and givens

On R∖{4}, f(x)=(-8)x+(-5). Find its punctured two-sided limit as x→4, without assigning f(4).


- G001 — On R∖{4}, f(x)=(-8)x+(-5). Find its punctured two-sided limit as x→4, without assigning f(4).





## 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

source roles = [qf-function-limit-affine, affine_punctured_limit, punctured_reals, []]

After

derived = [[integer_approach_value_STRING_OMIT1]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependenciesRule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001





S002 · calculationBefore

Punctured affine ORIGINAL slope times approach center := mul(-8, 4)

After

Punctured affine ORIGINAL slope times approach center = -32

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

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





S003 · calculationBefore

Punctured affine ORIGINAL approach affine constant sum := sum(-32, -5)

After

Punctured affine ORIGINAL approach affine constant sum = -37

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

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





S004 · calculationBefore

Punctured affine ORIGINAL positive slope bound one plus abs m := sum(1, 8)

After

Punctured affine ORIGINAL positive slope bound one plus abs m = 9

Conditions: none

Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.

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





S005 · impliesBefore

source roles = [Punctured affine, -8, -5, 4]

After

derived = [-37, 9, error = m*(x-a), delta = epsilon/9]

Conditions: none

For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof.

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









## Reverse checks


- S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S004 · premise_and_result_audit — Recompute Punctured affine ORIGINAL positive slope bound one plus abs m with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute Punctured affine ORIGINAL approach affine constant sum with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Punctured affine ORIGINAL slope times approach center with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.





## Result

-37

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: b41a57e6aae4d0b6afe26454568760f6376962823f904087c4fbcf032f6fe487
- candidate record SHA-256: 19e47cc8ec3c8b5b8fad9aabeecf9937d33e4f72ea0b938d0e2a5b02a17cb13b
- semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
- question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r02-function-limit-0008.json)