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Limit laws with given limits

Problem 1.506 · easy

Suppose \( \displaystyle \lim_{x\to -2} f(x) = 4 \), \( \displaystyle \lim_{x\to -2} g(x) = 1 \) and \( \displaystyle \lim_{x\to -2} h(x) = -5 \). Find \( \displaystyle \lim_{x\to -2} \left(f(x) g(x) - h(x)\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (product and difference) let us replace every function by its limit.
Reviewed
\[ 4 \cdot 1 + 5 = 9 \]
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: Line 1 is a sentence, not an algebraic equivalence. It correctly says the limits are finite, so the product and difference laws apply. The mention of denominators is harmless, since there are none.
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 +5 for h(x) instead of -5, violating the given limit lim h(x) = -5. The expression is f(x)g(x) - h(x), so it should be 4*1 - (-5) = 9, but the text claims to substitute the limits directly into the expression without accounting for the sign change in the subtraction, or simply performs the arithmetic 4*1 + 5 implying h(x) was 5 or the operation was addition. Wait, 4*1 - (-5) is 9. The equation line says 4*1 + 5 = 9. This is arithmetically correct for the final answer, but the sentence says 'replace every function by its limit'. If you replace h(x) by -5, you get 4*1 - (-5). The equation line writes 4*1 + 5. This is a subtle error in the explanation: it implies substituting h(x) with 5 or changing the operation. A student might think lim h(x) is 5. The setup is misleading because it obscures the sign handling.

Senior review claude-sonnet-5-5, 2026-10-10: pass — The limit laws for products and differences apply, giving 4*1 - (-5) = 9. The line 4*1 + 5 = 9 is the same value with the double negative resolved, and the stated answer is correct. Line 1 is a sentence that correctly justifies using the limit laws.

qwen3.6:27b-mlx: dismiss — The line 4*1 + 5 is the correct result of substituting the limits into f*g - h, since -h(x) tends to -(-5) = +5. Substituting h = -5 gives 4*1 - (-5) = 4*1 + 5 = 9, so there is no sign error and the answer is not a coincidence.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution incorrectly substitutes +5 for h(x) instead of -5, violating the given limit lim h(x) = -5. The expression is f(x)g(x) - h(x), so it should be 4*1 - (-5) = 9, but the text claims to substitute the limits directly into the expression without accounting for the sign change in the subtraction, or simply performs the arithmetic 4*1 + 5 implying h(x) was 5 or the operation was addition. Wait, 4*1 - (-5) is 9. The equation line says 4*1 + 5 = 9. This is arithmetically correct for the final answer, but the sentence says 'replace every function by its limit'. If you replace h(x) by -5, you get 4*1 - (-5). The equation line writes 4*1 + 5. This is a subtle error in the explanation: it implies substituting h(x) with 5 or changing the operation. A student might think lim h(x) is 5. The setup is misleading because it obscures the sign handling.
claude-sonnet-5-5: pass 2026-10-10 — The limit laws for products and differences apply, giving 4*1 - (-5) = 9. The line 4*1 + 5 = 9 is the same value with the double negative resolved, and the stated answer is correct. Line 1 is a sentence that correctly justifies using the limit laws.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution incorrectly substitutes +5 for h(x) instead of -5, leading to the wrong arithmetic result. The limit of h(x) is -5, so the expression should evaluate to 4*1 - (-5) = 9, but the equation line shows 4*1 + 5, implying the student treated the term as -h(x) where h(x)=5 or simply made a sign error in substitution while getting the right answer by coincidence or bad logic.

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 2026-10-10 with SymPy 1.14.0.