Home›Calculus 1›Limit laws with given limits›Problem 1.334

Limit laws with given limits

Problem 1.334 · easy

Suppose \( \displaystyle \lim_{x\to 0} f(x) = 8 \), \( \displaystyle \lim_{x\to 0} g(x) = 8 \) and \( \displaystyle \lim_{x\to 0} h(x) = -2 \). Find \( \displaystyle \lim_{x\to 0} \left(\left(f(x) - g(x)\right)^{2}\right) \).

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
\[ \left(-8 + 8\right)^{2} = 0 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 0 \)

✓ 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 sentence, not an equation. It correctly says the limit laws (difference, power) apply because f and g have finite limits. The remark about denominators is irrelevant but harmless, and h is unused.
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 uses -8 for f(x) instead of the given limit 8. While the final answer 0 is coincidentally correct, the reasoning contains a factual error in applying the given values.

Senior review claude-sonnet-5-5, 2026-10-06: pass — The answer 0 is correct and every stated equality holds. The only blemishes are cosmetic: the substitution is written as (-8+8) instead of (8-8), it mentions 'three limits' although h plays no part, and it refers to denominators when there are none.

qwen3.6:27b-mlx: dismiss — (-8+8)**2 is just (f-g)**2 with the limits substituted and the terms reordered as (-g+f), so it is not a wrong substitution; the value 0 is correct and the equation holds.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The substitution in step 2 is incorrect: it uses -8 for f(x) instead of the given limit 8. While the final answer 0 is coincidentally correct, the reasoning contains a factual error in applying the given values.
claude-sonnet-5-5: pass 2026-10-06 — The answer 0 is correct and every stated equality holds. The only blemishes are cosmetic: the substitution is written as (-8+8) instead of (8-8), it mentions 'three limits' although h plays no part, and it refers to denominators when there are none.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly substitutes -8 for the limit of f(x), which is given as 8. This arithmetic error leads to the wrong intermediate value, although the final answer happens to be correct by coincidence.

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 2026-10-06 with SymPy 1.14.0.