Question 23604: Both Sides of the Interior Singularity Need Independent Limits

Original question and givens

Use c=-9, interval [c−1,c+1], left integrand=4*(c−x)^(−3/4) and right integrand=5*(x−c)^(−0/1). 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=-9, interval [c−1,c+1], left integrand=4*(c−x)^(−3/4) and right integrand=5*(x−c)^(−0/1). 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(4, 3)

Improper LEFT reduced exponent numerator gap = 1

Improper LEFT reciprocal of one minus exponent := div(4, 1)

Improper LEFT reciprocal of one minus exponent = 4

Improper LEFT scale times singular-side integral := mul(4, 4)

Improper LEFT scale times singular-side integral = 16

ORIGINAL LEFT side at c=-9;scale=4;REDUCED exponent=3/4 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=16

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

Improper RIGHT reduced exponent numerator gap = 1

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

Improper RIGHT reciprocal of one minus exponent = 1

Improper RIGHT scale times singular-side integral := mul(5, 1)

Improper RIGHT scale times singular-side integral = 5

ORIGINAL RIGHT side at c=-9;scale=5;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=5

Improper independent singular-side limits sum := sum(16, 5)

Improper independent singular-side limits sum = 21

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

([-10,-8], -9, left:4*(c-x)^(-3/4), right:5*(x-c)^(-0/1), 16, 5, Yes, Yes, 21, 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.