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Limit laws with given limits

Problem 1.778 · easy

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

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (difference and quotient) let us replace every function by its limit.
Reviewed
\[ - \frac{1}{2} + 2 \cdot 4 = \frac{15}{2} \]
Substitute the three limits and simplify.✓ Proved

Answer \( \frac{15}{2} \)

Lines: 1 proved, 1 reviewed. 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies limit laws and performs the arithmetic accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies limit laws and performs the arithmetic accurately.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly applies limit laws for sums, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: inconclusive 2026-10-30 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution in line 2 is arithmetically incorrect. The expression is 4(2) - 1/2 = 7.5, but the line calculates -1/2 + 8 = 7.5? No, wait. 4*2 =
gpt-oss:20b: pass 2026-10-30

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 2026-10-31 with SymPy 1.14.0.