Question 23621: Both Sides of the Interior Singularity Need Independent Limits

Original question and givens

Use c=8, interval [c−1,c+1], left integrand=21*(c−x)^(−0/1) and right integrand=22*(x−c)^(−1/2). Both exponents are reduced rationals in [0,1). Give (interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate).

• G001 — Use c=8, interval [c−1,c+1], left integrand=21*(c−x)^(−0/1) and right integrand=22*(x−c)^(−1/2). Both exponents are reduced rationals in [0,1). Give (interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate).

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-improper-integral

Step-by-step derivation

source operation=improper-interior-singular-split;original snapshot and mathematics retained

explicit ordered source output fields=[interval, singularity, left_integrand, right_integrand, left_integral, right_integral, left_converges, right_converges, sum, certificate]

Improper LEFT reduced exponent numerator gap := sub(1, 0)

Improper LEFT reduced exponent numerator gap = 1

Improper LEFT reciprocal of one minus exponent := div(1, 1)

Improper LEFT reciprocal of one minus exponent = 1

Improper LEFT scale times singular-side integral := mul(21, 1)

Improper LEFT scale times singular-side integral = 21

ORIGINAL LEFT side at c=8;scale=21;REDUCED exponent=0/1 in[0,1)

put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=21

Improper RIGHT reduced exponent numerator gap := sub(2, 1)

Improper RIGHT reduced exponent numerator gap = 1

Improper RIGHT reciprocal of one minus exponent := div(2, 1)

Improper RIGHT reciprocal of one minus exponent = 2

Improper RIGHT scale times singular-side integral := mul(22, 2)

Improper RIGHT scale times singular-side integral = 44

ORIGINAL RIGHT side at c=8;scale=22;REDUCED exponent=1/2 in[0,1)

put distance t>0 from c;integral_delta^1 scale*t^(-exponent)=scale/(1-exponent)*(1-delta^(1-exponent));positive1-exponent makes delta term vanish,side limit=44

Improper independent singular-side limits sum := sum(21, 44)

Improper independent singular-side limits sum = 65

Reverse checks

• S010 · premise_and_result_audit — Recompute Improper independent singular-side limits sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute Improper RIGHT scale times singular-side integral with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Improper RIGHT reciprocal of one minus exponent with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Improper RIGHT reduced exponent numerator gap with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute Improper LEFT scale times singular-side integral with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Improper LEFT reciprocal of one minus exponent with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Improper LEFT reduced exponent numerator gap 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

([7,9], 8, left:21*(c-x)^(-0/1), right:22*(x-c)^(-1/2), 21, 44, Yes, Yes, 65, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)

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.