# Question 32326: Two-Sided Limit of Affine Branches

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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.

R0063-32326 · q-20261007-advance-20261008-r02-function-limit-0076



## Original question and givens

On the punctured domain R∖{-12}, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). Find the finite two-sided limit, or answer does not exist when the two side limits differ.


- G001 — On the punctured domain R∖{-12}, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). Find the finite two-sided limit, or answer does not exist when the two side limits differ.





## 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-two-sided-affine, two_sided_affine_limit, punctured_reals, []]

After

derived = [[integer_approach_value_OR_source_nonexistence_STRING]]

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

LEFT affine approach ORIGINAL slope times approach center := mul(1, -12)

After

LEFT affine approach ORIGINAL slope times approach center = -12

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

LEFT affine approach ORIGINAL approach affine constant sum := sum(-12, 0)

After

LEFT affine approach ORIGINAL approach affine constant sum = -12

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

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

After

LEFT affine approach ORIGINAL positive slope bound one plus abs m = 2

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 = [LEFT affine approach, 1, 0, -12]

After

derived = [-12, 2, error = m*(x-a), delta = epsilon/2]

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





S006 · calculationBefore

RIGHT affine approach ORIGINAL slope times approach center := mul(-1, -12)

After

RIGHT affine approach ORIGINAL slope times approach center = 12

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, S005





S007 · calculationBefore

RIGHT affine approach ORIGINAL approach affine constant sum := sum(12, -24)

After

RIGHT affine approach ORIGINAL approach affine constant sum = -12

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, S006





S008 · calculationBefore

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

After

RIGHT affine approach ORIGINAL positive slope bound one plus abs m = 2

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, S007





S009 · impliesBefore

source roles = [RIGHT affine approach, -1, -24, -12]

After

derived = [-12, 2, error = m*(x-a), delta = epsilon/2]

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, S008





S010 · calculationBefore

Two-sided ORIGINAL positive joint slope bound := sum(1, 1)

After

Two-sided ORIGINAL positive joint slope bound = 2

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, S009





S011 · impliesBefore

source roles = [-12, 1, 0, -1, -24]

After

derived = [-12, -12, 2, delta = epsilon/2]

Conditions: none

When both approach values AGREE,δ=ε/(1+max(|lm|,|rm|))controls BOTH sides for ANY ε>0,so their common value is the two-sided limit,including both slopes0.No original f(a)is assigned.

Step identifiers and dependenciesRule: Limit_TWO_sides_AGREE_shared_delta_whole_punctured_law · Uses: G001, S010









## Reverse checks


- S011 · 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.
- S010 · premise_and_result_audit — Recompute Two-sided ORIGINAL positive joint slope bound with exact rational arithmetic, not rounded decimals.
- S009 · 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.
- S008 · premise_and_result_audit — Recompute RIGHT affine approach ORIGINAL positive slope bound one plus abs m with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute RIGHT affine approach ORIGINAL approach affine constant sum with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute RIGHT affine approach ORIGINAL slope times approach center with exact rational arithmetic, not rounded decimals.
- 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 LEFT affine approach ORIGINAL positive slope bound one plus abs m with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute LEFT affine approach ORIGINAL approach affine constant sum with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute LEFT affine approach 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

-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: 5ab3924139bbbe15e47a33808cd5c8bbdd93df9e617ab2e693547433fb2f8af9
- candidate record SHA-256: a6c9f2360da8ac01544340b07d40207b591debe7e38b04e9d179d7b79823f6af
- 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-0076.json)