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Limit laws with given limits

Problem 1.439 · easy

Suppose \( \displaystyle \lim_{x\to 5} f(x) = 3 \), \( \displaystyle \lim_{x\to 5} g(x) = 7 \) and \( \displaystyle \lim_{x\to 5} h(x) = -5 \). Find \( \displaystyle \lim_{x\to 5} \left(f(x) g(x) - h(x)\right) \).

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
\[ 5 + 3 \cdot 7 = 26 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 26 \)

✓ 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 invoking the product and difference limit laws, which apply because each of the three limits is finite. No denominators occur, so the remark about denominators is vacuous but harmless. The justification is legitimate.
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 the limit of f(x), which is given as 3. The correct calculation should be 3*7 - (-5) = 26, but the equation shown is 5 + 3*7, implying f(x) -> 5 and a sign error in the subtraction.

Senior review claude-sonnet-5-5, 2026-10-08: pass — The arithmetic is correct and the answer 26 is right. The line skips showing the intermediate 3*7 - (-5), so a reader may not see where the +5 comes from, but this is not a mathematical error.

qwen3.6:27b-mlx: dismiss — The line 5 + 3*7 = 26 is the correct result of 3*7 - (-5), since subtracting -5 is adding 5; the 5 is the negated limit of h, not a substitution of h -> 5, and SymPy confirms the equation is true.

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 5 for the limit of f(x), which is given as 3. The correct calculation should be 3*7 - (-5) = 26, but the equation shown is 5 + 3*7, implying f(x) -> 5 and a sign error in the subtraction.
claude-sonnet-5-5: pass 2026-10-08 — The arithmetic is correct and the answer 26 is right. The line skips showing the intermediate 3*7 - (-5), so a reader may not see where the +5 comes from, but this is not a mathematical error.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution substitutes 5 for h(x) instead of the given limit -5. The correct calculation is 3*7 - (-5) = 26, but the equation shown is 5 + 3*7 = 26, which implies h(x) -> 5 and uses addition instead of subtraction.

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.