# Question 28753: Limit of an Explicit Reciprocal-Tail Sequence

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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.

R0056-28753 · q-20261007-advance-20261008-r01-sequence-limit-0003



## Original question and givens

On natural indices starting at zero, u(n)=(-10)+(3)/(n+3), with d≥1. Find the declared reciprocal-tail limit.


- G001 — On natural indices starting at zero, u(n)=(-10)+(3)/(n+3), with d≥1. Find the declared reciprocal-tail limit.





## 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-sequence-limit](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001





S002 · impliesBefore

source roles = [3, 3, naturals_from_zero]

After

derived = [3, 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.

Step identifiers and dependenciesRule: Sequence_WHOLE_law_and_positive_increasing_denominator · Uses: G001, S001





S003 · impliesBefore

source roles = [-10, 3, 3]

After

derived = [-10, 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.

Step identifiers and dependenciesRule: Sequence_limit_EVERY_epsilon_has_a_whole_tail · Uses: G001, S002









## Reverse checks


- 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.





## Result

-10

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: af45aeb70640dad44943b6e99e6b0261eb4c122f6d408151137c064ff1b0a520
- candidate record SHA-256: f78d7b89236776e6667df0257e21199971cea76efcd93b9f9df39888dc813f91
- semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8
- question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r01-sequence-limit-0003.json)