Home›Calculus 1›Limit laws with given limits›Problem 1.333

Limit laws with given limits

Problem 1.333 · easy

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

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (quotient of a sum) let us replace every function by its limit.
Reviewed
\[ 1 \frac{1}{-3 + 2} = -1 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -1 \)

✓ 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 algebraic equivalence. Its claims are true: all three limits are finite, and the denominator g+h has limit 2+(-3) = -1, which is nonzero. The Sum and Quotient Laws therefore justify substituting the limits.
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 the limit laws for quotients and sums, verifying that the denominator's limit is non-zero.

Senior review claude-sonnet-5-5, 2026-10-06: pass — The limit laws apply because f, g and h have finite limits and the denominator's limit 2+(-3) = -1 is nonzero. The result 1/(-1) = -1 matches the stated answer.

qwen3.6:27b-mlx: dismiss — 'No denominator's limit is 0' refers to the quotient's denominator g+h, whose limit -1 is nonzero, so the statement is true; 'quotient of a sum' loosely but accurately describes applying the Quotient and Sum Laws, and this is at most a style quibble.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the limit laws for quotients and sums, verifying that the denominator's limit is non-zero.
claude-sonnet-5-5: pass 2026-10-06 — The limit laws apply because f, g and h have finite limits and the denominator's limit 2+(-3) = -1 is nonzero. The result 1/(-1) = -1 matches the stated answer.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The sentence claims 'no denominator's limit is 0', but the denominator is g(x) + h(x), whose limit is 2 + (-3) = -1. While -1 is not 0, the phrasing implies checking individual functions g and h for zero limits, which is irrelevant. More importantly, the phrase 'quotient of a sum' is not a standard name for the limit laws applied here (Sum Law then Quotient Law), and the explanation is confusingly imprecise.

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.