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Limit laws with given limits

Problem 1.777 · easy

Suppose \( \displaystyle \lim_{x\to 6} f(x) = 8 \), \( \displaystyle \lim_{x\to 6} g(x) = -3 \) and \( \displaystyle \lim_{x\to 6} h(x) = 3 \). Find \( \displaystyle \lim_{x\to 6} \left(4 f(x) - \frac{g(x)}{h(x)}\right) \).

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.
\[ \frac{3}{3} + 8 \cdot 4 = 33 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 33 \)

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. How we verify

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 applies the limit laws by substituting the limits into an expression that does not match the problem statement. The problem asks for the limit of 4f(x) - g(x)/h(x), but the equation computes 3/3 + 8*4, which corresponds to g(x)/h(x) + 4f(x). The sign of the quotient term is flipped (addition instead of subtraction), leading to the wrong result (33 instead of 29).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution incorrectly applies the limit laws by substituting the limits into an expression that does not match the problem statement. The problem asks for the limit of 4f(x) - g(x)/h(x), but the equation computes 3/3 + 8*4, which corresponds to g(x)/h(x) + 4f(x). The sign of the quotient term is flipped (addition instead of subtraction), leading to the wrong result (33 instead of 29).
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The solution incorrectly applies the limit laws by substituting the limits into an expression that does not match the problem statement (calculating 3/3 + 8*4 instead of 4*8 - (-3)/3). This results in the wrong arithmetic setup and a misleading demonstration of how to apply limit laws.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly applies limit laws for sums, differences, products, and quotients, as the denominator's limit is non-zero. The arithmetic is correct.
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.