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Limit laws with given limits

Problem 1.214 · easy

Suppose \( \displaystyle \lim_{x\to 4} f(x) = 3 \), \( \displaystyle \lim_{x\to 4} g(x) = -5 \) and \( \displaystyle \lim_{x\to 4} h(x) = -4 \). Find \( \displaystyle \lim_{x\to 4} \left(2 f(x) + 3 g(x)\right) \).

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
\[ \left(-5\right) 3 + 3 \cdot 2 = -9 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -9 \)

✓ 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 that invokes the sum and constant-multiple limit laws, which legitimately apply because each limit is finite. The mention of denominators 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: fail (error) — The solution substitutes the wrong values for the functions. It uses 3 for g(x) and 2 for f(x), whereas the problem states lim f(x) = 3 and lim g(x) = -5. The correct calculation should be 2(3) + 3(-5) = -9, but the equation shown (-5*3 + 3*2) implies incorrect substitutions or a scrambled expression.

Senior review claude-sonnet-5-5, 2026-10-04: pass — The final value -9 is correct, and the displayed arithmetic -15 + 6 = -9 is a valid commuted form of 2(3) + 3(-5). The remark about denominators in line 1 is vacuous, since there are none, and h is unused, but neither is a defect.

qwen3.6:27b-mlx: dismiss — The terms are just reordered: -5*3 is lim g times its coefficient 3, and 3*2 is lim f times its coefficient 2. This equals 2*3 + 3*(-5) = -9, so no wrong values are substituted.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution substitutes the wrong values for the functions. It uses 3 for g(x) and 2 for f(x), whereas the problem states lim f(x) = 3 and lim g(x) = -5. The correct calculation should be 2(3) + 3(-5) = -9, but the equation shown (-5*3 + 3*2) implies incorrect substitutions or a scrambled expression.
gpt-oss:20b: pass 2026-10-04
claude-sonnet-5-5: pass 2026-10-04 — The final value -9 is correct, and the displayed arithmetic -15 + 6 = -9 is a valid commuted form of 2(3) + 3(-5). The remark about denominators in line 1 is vacuous, since there are none, and h is unused, but neither is a defect.
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution substitutes the wrong values for the functions. It uses 3 for g(x) and 2 for f(x), whereas the problem states lim f(x) = 3 and lim g(x) = -5. The correct calculation should be 2(3) + 3(-5) = 6 - 15 = -9, but the equation shown (-5*3 + 3*2) implies lim f(x)=2 and lim g(x)=3, which contradicts the problem statement.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.