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R0056-28759 · q-20261007-advance-20261008-r01-sequence-limit-0009
On natural indices starting at zero, u(n)=(-4)+(9)/(n+4), with d≥1. Find the declared reciprocal-tail limit.
G001 — On natural indices starting at zero, u(n)=(-4)+(9)/(n+4), with d≥1. Find the declared reciprocal-tail limit.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.
Before
source roles = [qf-sequence-limit-reciprocal-value, reciprocal_sequence_limit]
After
derived = [limit_from_whole_reciprocal_sequence_law]
Conditions: none
Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
source roles = [9, 4, naturals_from_zero]
After
derived = [9, u(n)=L+c/(n+d), |u(n)-L|=|c|/(n+d)]
Conditions: none
The premise is the WHOLE law for every n>=0,not a finite prefix.With d>0,the denominator is positive and increasing.Subtract L before taking the absolute value.In error/cutoff tasks L is an arbitrary fixed real,not an invented input parameter.
Rule: Sequence_WHOLE_law_and_positive_increasing_denominator · Uses: G001, S001
Before
source roles = [-4, 9, 4]
After
derived = [-4, N=max(0,floor(|c|/epsilon)-d+1), for all n>=N: |u(n)-L|<epsilon]
Conditions: none
For ANY epsilon>0 choose N=max(0,floor(|c|/epsilon)-d+1).Then n+d>|c|/epsilon for EVERY n>=N,so the error is strictly below epsilon.For c=0,N=0 suffices.Thus the limit is the GIVEN L,not a sampled term.
Rule: Sequence_limit_EVERY_epsilon_has_a_whole_tail · Uses: G001, S002
S003 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S002 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.-4
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 3bc137b2508a5b4626e0ba81690e23988fa32fa3d41eecceec24afaa681b08f6candidate record SHA-256: bd64ff7ce5d0cd1b8f5dc46d5ab141c3ea29480c676e3f90485116b206ffea0esemantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED