OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 3911: Ordered Center-Value and Punctured-Limit Pair

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 20

← Question derivation index · Original question batch

R0006-03911 · q-20261007-advance-20261009-r03-function-limit-0036

Original question and givens

The full-real function has affine expression (-2)x+(-6) away from a=-2, and separately defined center value -1. Give (center value,punctured two-sided limit).

  • G001 — The full-real function has affine expression (-2)x+(-6) away from a=-2, and separately defined center value -1. Give (center value,punctured two-sided 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.

Step-by-step derivation

S001 · implies

Before

declared real model with an excluded center for limit data

After

point value,legal off-center evaluation and limit are distinct

Conditions: none

A punctured limit does not read the center value,and evaluation at one legal point is not a limit.These declared affine or factorizable models do not represent arbitrary functions.

Step identifiers and dependencies

Rule: punctured_limit_scope · Uses: G001

S002 · calculation

Before

punctured branch slope times center := mul(-2, -2)

After

punctured branch slope times center = 4

Conditions: none

Multiply the branch slope by the center.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

punctured branch limit value := sum(4, -6)

After

punctured branch limit value = -2

Conditions: none

Add that branch's constant;the affine difference m(x−a) admits ε−δ control.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

given f(a)=-1;punctured affine limit=-2

After

ordered pair=(-1,-2)

Conditions: none

The center value is given separately.Off-center f(x)−L=m(x−a),so δ=ε/(1+|m|) proves the limit without changing the point value or assuming continuity.

Step identifiers and dependencies

Rule: point_value_independent_of_limit · Uses: G001, S003

Reverse checks

  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute punctured branch limit value with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute punctured 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

(-1, -2)

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: 8cfd78e99419ee93c49174b073d2d59b803a234241c629e14bdfe8b6fadd32aa
  • candidate record SHA-256: d7a4a93744e28392ffa699713c4ecba8a458cb3e894b8c55af0d7f778006c740
  • semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
  • question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON