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Limit laws with given limits

Problem 1.433 · easy

Suppose \( \displaystyle \lim_{x\to 6} f(x) = 3 \), \( \displaystyle \lim_{x\to 6} g(x) = 9 \) and \( \displaystyle \lim_{x\to 6} h(x) = -1 \). Find \( \displaystyle \lim_{x\to 6} \left(\left(f(x) - g(x)\right)^{2}\right) \).

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
\[ \left(-9 + 3\right)^{2} = 36 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 36 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is a prose justification, not an algebraic equivalence. The difference law and power law apply because f and g have finite limits, and there are no denominators. The extra limit of h is irrelevant but harmless.
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution in step 2 is incorrect; it calculates (3 - 9)^2 = 36, but the expression is (f(x) - g(x))^2, so it should be (3 - 9)^2. Wait, 3-9 i

Senior review claude-sonnet-5-5, 2026-10-08: pass — Substituting the limits gives (3−9)² = (−6)² = 36, which matches the stated answer. The line 1 sentence is a valid appeal to the limit laws (the denominator clause holds vacuously, and h is unused).

qwen3.6:27b-mlx: dismiss — The limit of f−g is 3−9, which equals −9+3 = −6, so (−9+3)² is exactly the substitution for (f−g)²; the objector confused −9+3 with 9−3.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution in step 2 is incorrect; it calculates (3 - 9)^2 = 36, but the expression is (f(x) - g(x))^2, so it should be (3 - 9)^2. Wait, 3-9 i
claude-sonnet-5-5: pass 2026-10-08 — Substituting the limits gives (3−9)² = (−6)² = 36, which matches the stated answer. The line 1 sentence is a valid appeal to the limit laws (the denominator clause holds vacuously, and h is unused).
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The substitution in step 2 is incorrect: it calculates (-9 + 3)^2, which corresponds to (g(x) - f(x))^2, but the problem asks for (f(x) - g(x))^2. Although the final numerical answer happens to be correct because the expression is squared, the intermediate step demonstrates a sign error in applying the limit laws.

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 2026-10-08 with SymPy 1.14.0.