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Limit laws with given limits

Problem 1.217 · easy

Suppose \( \displaystyle \lim_{x\to 4} f(x) = -6 \), \( \displaystyle \lim_{x\to 4} g(x) = 8 \) and \( \displaystyle \lim_{x\to 4} h(x) = 2 \). Find \( \displaystyle \lim_{x\to 4} \left(\left(f(x) - g(x)\right)^{2}\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (power) let us replace every function by its limit.
Reviewed
\[ \left(-8 - 6\right)^{2} = 196 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 196 \)

Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: Line 1 is a sentence, not an equation. It justifies applying the limit laws (difference and power) using the finite limits of f and g; h is irrelevant to the expression and there is no denominator. The claim is true and no algebraic script applies.
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the limit of the same combination of concrete functions with those limits, taken by SymPy

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The substitution in step 2 is incorrect: it computes (-8 - 6)^2, implying f(4)=-8 and g(4)=6, but the problem states lim f(x)=-6 and lim g(x)=8. The correct calculation should be (-6 - 8)^2 = 196. Although the final answer is coincidentally correct, the intermediate step demonstrates a fundamental error in applying the given values.

Senior review claude-sonnet-5-5, 2026-10-04: fail — The answer 196 is correct, but line 2 claims to substitute the limits and writes (-8-6)^2 instead of (-6-8)^2, which misplaces the values for f and g. A student could take this as f→-8 and g→6 or as a sign slip, so the step should read (-6 - 8)**2 = 196.

qwen3.6:27b-mlx: uphold — Substituting f→-6 and g→8 into (f-g)^2 gives (-6-8)^2, not (-8-6)^2; the written expression happens to equal -14 but does not follow from the stated substitution, and the answer 196 is right only because of the squaring.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The substitution in step 2 is incorrect: it computes (-8 - 6)^2, implying f(4)=-8 and g(4)=6, but the problem states lim f(x)=-6 and lim g(x)=8. The correct calculation should be (-6 - 8)^2 = 196. Although the final answer is coincidentally correct, the intermediate step demonstrates a fundamental error in applying the given values.
gpt-oss:20b: pass 2026-10-04
claude-sonnet-5-5: fail (misleading) 2026-10-04 — The answer 196 is correct, but line 2 claims to substitute the limits and writes (-8-6)^2 instead of (-6-8)^2, which misplaces the values for f and g. A student could take this as f→-8 and g→6 or as a sign slip, so the step should read (-6 - 8)**2 = 196.
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The substitution in step 2 is incorrect: it calculates (-8 - 6)^2, implying f(x) -> 8 and g(x) -> -6, whereas the problem states f(x) -> -6 and g(x) -> 8. The correct calculation should be (-6 - 8)^2 = 196. Although the final numerical answer is coincidentally correct due to the square, the intermediate step demonstrates a fundamental error in applying the given limits.
gpt-oss:20b: pass 2026-10-04

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/limit_laws_values, checked 2026-10-04 with SymPy 1.14.0.