On R∖{1}, f(x)=(21)x+(8). Find its punctured two-sided limit as x→1, without assigning f(1).
G001 — On R∖{1}, f(x)=(21)x+(8). Find its punctured two-sided limit as x→1, without assigning f(1).
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.
Punctured affine ORIGINAL slope times approach center := mul(21, 1)
After
Punctured affine ORIGINAL slope times approach center = 21
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.
Punctured affine ORIGINAL approach affine constant sum := sum(21, 8)
After
Punctured affine ORIGINAL approach affine constant sum = 29
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.
Punctured affine ORIGINAL positive slope bound one plus abs m := sum(1, 21)
After
Punctured affine ORIGINAL positive slope bound one plus abs m = 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.
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.
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
29
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.