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Limit laws with given limits

Problem 1.560 · easy

Suppose \( \displaystyle \lim_{x\to 2} f(x) = 4 \), \( \displaystyle \lim_{x\to 2} g(x) = 1 \) and \( \displaystyle \lim_{x\to 2} h(x) = -3 \). Find \( \displaystyle \lim_{x\to 2} \left(4 f(x) - \frac{g(x)}{h(x)}\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (difference and quotient) let us replace every function by its limit.
Reviewed
\[ - \frac{1}{-3} + 4 \cdot 4 = \frac{49}{3} \]
Substitute the three limits and simplify.✓ Proved

Answer \( \frac{49}{3} \)

✓ 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 sentence, not an equation. It correctly says the limit laws apply: each limit is finite and the denominator's limit, -3, is nonzero. The constant-multiple law for 4f is implicit and 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: pass — The solution correctly applies limit laws for sums, differences, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The limit laws apply because lim h = -3 is nonzero. Substituting gives 16 - (1/(-3)) = 49/3, which matches the stated answer. The step omits the constant-multiple law, which is a trivial gap.

gpt-oss:20b: dismiss — The limit is 4*4 - (1/(-3)) = 16 + 1/3 = 49/3; the objection forgets the subtraction sign, which turns -1/3 into +1/3, so 47/3 is wrong.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies limit laws for sums, differences, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.
claude-sonnet-5-5: pass 2026-10-12 — The limit laws apply because lim h = -3 is nonzero. Substituting gives 16 - (1/(-3)) = 49/3, which matches the stated answer. The step omits the constant-multiple law, which is a trivial gap.
gpt-oss:20b: fail (error) 2026-10-12 — The quotient term is evaluated incorrectly: g(x)/h(x) tends to 1/(-3) = -1/3, not +1/3. Thus the correct limit is 47/3, not 49/3.
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies limit laws for sums, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.

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 2026-10-12 with SymPy 1.14.0.