On R∖{11}, f(x)=(x²−(11)²)/(x−(11)). Find its punctured two-sided limit as x→11.
G001 — On R∖{11}, f(x)=(x²−(11)²)/(x−(11)). Find its punctured two-sided limit as x→11.
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.
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.
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.
Removable ORIGINAL negative a squared constant := sub(0, 121)
After
Removable ORIGINAL negative a squared constant = -121
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.
Removable ORIGINAL cross terms a minus a := sub(11, 11)
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.
Removable ORIGINAL approach a plus a := sum(11, 11)
After
Removable ORIGINAL approach a plus a = 22
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.
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.
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.
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
22
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.