Home›Calculus 1›Limit laws with given limits›Problem 1.431

Limit laws with given limits

Problem 1.431 · easy

Suppose \( \displaystyle \lim_{x\to 1} f(x) = 2 \), \( \displaystyle \lim_{x\to 1} g(x) = 9 \) and \( \displaystyle \lim_{x\to 1} h(x) = 4 \). Find \( \displaystyle \lim_{x\to 1} \left(2 f(x) + 3 g(x)\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (sum and constant multiple) let us replace every function by its limit.
Reviewed
\[ 2 \cdot 2 + 9 \cdot 3 = 31 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 31 \)

✓ 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: Sentence: the sum and constant multiple limit laws apply because f and g have finite limits; the remark about denominators is vacuously true as no quotient appears.
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 solution incorrectly substitutes 9 for g(x) in the term 3g(x), resulting in 9*3 instead of 3*9. While the arithmetic result is coincidentally correct, the substitution logic is flawed and misleading.

Senior review claude-sonnet-5-5, 2026-10-08: pass — The math is correct: 2*2 + 3*9 = 31. The mention of denominators is irrelevant boilerplate since there is no quotient, and h is a distractor, but nothing stated is false.

qwen3.6:27b-mlx: dismiss — 9*3 equals 3*9 by commutativity and the coefficient-to-limit mapping is correct (2 with f's limit 2, 3 with g's limit 9); the 'no denominator' remark is a harmless, true vacuous statement, not a defect.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly substitutes 9 for g(x) in the term 3g(x), resulting in 9*3 instead of 3*9. While the arithmetic result is coincidentally correct, the substitution logic is flawed and misleading.
claude-sonnet-5-5: pass 2026-10-08 — The math is correct: 2*2 + 3*9 = 31. The mention of denominators is irrelevant boilerplate since there is no quotient, and h is a distractor, but nothing stated is false.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly substitutes 9 for g(x) in the term 3g(x), resulting in 9*3 instead of 3*9 (though the arithmetic result is coincidentally correct, the mapping of coefficients to limits is described confusingly). More importantly, the sentence claims 'no denominator's limit is 0' which is irrelevant and misleading since there is no division in the expression.

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.