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Limit laws with given limits

Problem 1.564 · easy

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

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
\[ \left(-4\right) 3 - 4 \cdot 2 = -20 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -20 \)

✓ 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. The sum and constant-multiple limit laws apply because lim f and lim g are finite, so lim(2f+3g) = 2(-4)+3(-4) = -20. The mention of denominators and of h 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: pass — The solution correctly applies limit laws for sums and constant multiples. The calculation is accurate.

Senior review claude-sonnet-5-5, 2026-10-12: fail — The answer -20 is correct, and line 2 evaluates 3(-4)+2(-4) correctly. The text refers to 'three limits' and 'denominators' when only the f and g limits are used and no denominator exists, which is sloppy boilerplate but not false mathematics.

qwen3.6:27b-mlx: uphold — The arithmetic is correct (-12 - 8 = -20), but 'Substitute the three limits' is inaccurate because h does not appear in 2f+3g. This is a minor wording defect, not a math error, so it does not support the 'error' severity.

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 and constant multiples. The calculation is accurate.
claude-sonnet-5-5: fail (style) 2026-10-12 — The answer -20 is correct, and line 2 evaluates 3(-4)+2(-4) correctly. The text refers to 'three limits' and 'denominators' when only the f and g limits are used and no denominator exists, which is sloppy boilerplate but not false mathematics.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly substitutes the limit of h(x) into the expression for 2f(x) + 3g(x), which does not involve h(x). The correct calculation should be 2(-4) + 3(-4) = -20, but the reasoning implies using h(x) or mislabels the terms, leading to a logically flawed justification even if the arithmetic coincidence yields the right number. Specifically, line 1 claims to replace 'every function' but the expression only has f and g. Line 2 calculates -4*3 - 4*2, which corresponds to 3g + 2f, but the text says 'Substitute the three limits', implying h was used or relevant, which is false. More critically, the arithmetic in line 2 is -12 - 8 = -20, which is correct for 3(-4) + 2(-4). However, the sentence says 'Substitute the three limits', which is factually wrong as h is not in the expression. This is misleading/error in reasoning.

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.