# Question 32317: Punctured Limit of a Removable Quotient [Return to catalogue page 162](https://ocme.evemisslab.com/en/questions/page/162/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r02-function-limit-foundations/#q-20261007-advance-20261008-r02-function-limit-0042) 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. R0063-32317 · q-20261007-advance-20261008-r02-function-limit-0042 ## Original question and givens On R∖{4}, f(x)=(x²−(4)²)/(x−(4)). Find its punctured two-sided limit as x→4. - G001 — On R∖{4}, f(x)=(x²−(4)²)/(x−(4)). Find its punctured two-sided limit as x→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-removable-quotient, removable_quotient_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 Removable ORIGINAL a squared := mul(4, 4) After Removable ORIGINAL a squared = 16 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 Removable ORIGINAL negative a squared constant := sub(0, 16) After Removable ORIGINAL negative a squared constant = -16 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 Removable ORIGINAL cross terms a minus a := sub(4, 4) After Removable ORIGINAL cross terms a minus a = 0 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 · calculationBefore Removable ORIGINAL approach a plus a := sum(4, 4) After Removable ORIGINAL approach a plus a = 8 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, S004 S006 · impliesBefore source roles = [4] After derived = [-16, 0, 1, (x-a)*(x+a) = x^2-a^2, x-a != 0] Conditions: none Verify the COMPLETE coefficient identity(x−a)(x+a)=x²−a².ONLY x≠a in the punctured domain gives a nonzero denominator and permits cancellation to x+a.Never substitute0/0 at a or fill the original hole. Step identifiers and dependenciesRule: Limit_REMOVABLE_polynomial_factor_identity_CANCEL_ONLY_x_not_a · Uses: G001, S005 S007 · impliesBefore source roles = [4, 8] After derived = [(x+a)-2a = x-a, delta = epsilon, 8] Conditions: none With L=2a,the error is exactly|x−a|.For ANY ε>0 choose δ=ε;EVERY0<|x−a|<δ gives error<ε.This proves the punctured limit while the original quotient remains undefined at a. Step identifiers and dependenciesRule: Limit_REMOVABLE_punctured_error_delta_epsilon_NO_extension · Uses: G001, S006 ## Reverse checks - S007 · 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. - S006 · 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. - S005 · premise_and_result_audit — Recompute Removable ORIGINAL approach a plus a with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Removable ORIGINAL cross terms a minus a with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Removable ORIGINAL negative a squared constant with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Removable ORIGINAL a squared 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 8 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: 422ce7ef95efb4d55079f974aa3eb846ee7fe28c3a8cb07c032425ddcd422d81 - candidate record SHA-256: 984e837772727da8024392c0ce335d0c6eb48fcc86239abf6e03e654cbe5c0e3 - 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-0042.json)