# Question 3376: One-Sided Limit Defects Relative to the Point Value [Return to catalogue page 17](https://ocme.evemisslab.com/en/questions/page/17/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r03-continuity-foundations/#q-20261007-advance-20261008-r03-continuity-0076) 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. R0005-03376 · q-20261007-advance-20261008-r03-continuity-0076 ## Original question and givens On all R, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). The center value is -12. Give (left limit−center value, right limit−center value). - G001 — On all R, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). The center value is -12. Give (left limit−center value, right limit−center value). ## 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-continuity](https://ocme.evemisslab.com/en/mko/mko-continuity/) ## Step-by-step derivation S001 · impliesBefore declared real model;center a=-12 After distinguish punctured limits,point values and a new extension Conditions: none Use the declared real formulas;point values,punctured limits and values of a newly defined extension are different data. Step identifiers and dependenciesRule: continuity_model_scope · Uses: G001 S002 · calculationBefore left branch slope times center := mul(1, -12) After left branch slope times center = -12 Conditions: none Substitute the center a into this affine formula's product term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore left branch center value := sum(-12, 0) After left branch center value = -12 Conditions: none Then add this branch's intercept,not the other branch's constant. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore right branch slope times center := mul(-1, -12) After right branch slope times center = 12 Conditions: none Substitute the center a into this affine formula's product term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore right branch center value := sum(12, -24) After right branch center value = -12 Conditions: none Then add this branch's intercept,not the other branch's constant. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore left L=-12;right R=-12;given f(a)=-12 After these are one-sided limits,not automatic point substitutions into f(a) Conditions: none For an affine branch the difference from its center limit L is m(x−a).For any ε>0 choose δ=ε/(1+|m|) to obtain the one-sided limit.The actual center value remains the separately specified v. Step identifiers and dependenciesRule: affine_one_sided_limits · Uses: G001, S005 S007 · calculationBefore left limit minus point value := sub(-12, -12) After left limit minus point value = 0 Conditions: none Compute the left limit minus the actual point value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore right limit minus point value := sub(-12, -12) After right limit minus point value = 0 Conditions: none Compare the right limit with the same actual point value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore L−f(a)=0;R−f(a)=0 After continuous at a=Yes Conditions: none In this real model with affine one-sided limits,continuity holds exactly when both limits and the point value agree.Equal left and right limits alone may still leave an incorrect center value. Step identifiers and dependenciesRule: two_sided_limit_equals_point · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute right limit minus point value with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute left limit minus point value with exact rational arithmetic, not rounded decimals. - 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 center 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 center 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 (0, 0) 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: b8c4a351b2fea22f13807ac97d9218db6fa40a8cc22eef3728fbe8daa9281bef - candidate record SHA-256: e80a61d5d8593da45291f0a365234f316262c5c1fa834a5b3515ce0022935b58 - semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2 - question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r03-continuity-0076.json)