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Limit laws with given limits

Problem 1.338 · easy

Suppose \( \displaystyle \lim_{x\to 2} f(x) = -6 \), \( \displaystyle \lim_{x\to 2} g(x) = 5 \) and \( \displaystyle \lim_{x\to 2} h(x) = -2 \). 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.
\[ 2 - 6 \cdot 5 = -28 \]
Substitute the three limits and simplify.✓ Proved

Answer \( -28 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
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 the limit of h(x) as 2 instead of -2, leading to the wrong final answer. The correct calculation should be (-6)(5) - (-2) = -28 + 2 = -26.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly substitutes the limit of h(x) as 2 instead of -2, leading to the wrong final answer. The correct calculation should be (-6)(5) - (-2) = -28 + 2 = -26.
gpt-oss:20b: fail (error) 2026-10-06 — The calculation in step 2 uses "2 - 6*5" instead of the correct expression "(-6)*5 - (-2)". The limit should be computed as (-6)(5) - (-2) = -28, but the written expression is incorrect.
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The substitution in the equation is incorrect; it calculates 2 - 30 instead of (-6)(5) - (-2). The limit of h(x) is -2, so subtracting h(x) means subtracting -2, not 2.

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.