Home›Calculus 1›The squeeze theorem›Problem 1.734

The squeeze theorem

Problem 1.734 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\sin{\left(3 x \right)}}{x} \).

Since −1 ≤ sin(·) ≤ 1, the expression lies between -1/x and 1/x.
Reviewed
\[ \lim_{x \to \infty}\left(- \frac{1}{x}\right) = 0 \]
The lower bound's limit.✓ Proved
\[ \lim_{x \to \infty} \frac{1}{x} = 0 \]
The upper bound's limit, the same.✓ Proved
Both bounds tend to the same value, so the expression trapped between them does too.
Reviewed

Answer \( 0 \)

Lines: 2 proved, 2 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
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the expression evaluated ever closer to the limit point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the sine function and showing that both bounding functions converge to 0 as x approaches infinity.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the sine function and showing that both bounding functions converge to 0 as x approaches infinity.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly uses the bounds -1/x and 1/x for x approaching infinity. Since x is positive in this limit, dividing the inequality -1 <= s
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to multiply the bounds by 1/x, claiming the expression lies between -1/x and 1/x without showing the multiplication step. This skips the crucial algebraic manipulation required to set up the squeeze theorem correctly.
gpt-oss:20b: pass 2026-10-19

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/squeeze_theorem, checked 2026-10-20 with SymPy 1.14.0.