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The squeeze theorem

Problem 1.739 · medium

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

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 \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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. The bounds are valid for x > 0, and the limits of the bounding functions are correctly identified as 0.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the Squeeze Theorem. The bounds are valid for x > 0, and the limits of the bounding functions are correctly identified as 0.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for sin(x)/x and showing that the limits of these bounds are equal.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The solution fails to account for the sign of x. For x approaching positive infinity, 1/x is positive and -1/x is negative, so the inequality -1/x <= sin(x)/x <= 1/x holds. However, the squeeze theorem requires the bounds to sandwich the function. If the problem were x -> -infinity, the inequalities would reverse. While the specific limit x->oo makes the bounds correct, the sentence 'the expression lies between -1/x and 1/x' is technically imprecise without specifying x > 0, as for x < 0, 1/x < -1/x. More importantly, standard rigorous application requires stating the inequality for the specific domain (x > 0). A stricter review might flag this as 'misleading' or 'style', but given the adversarial nature, the lack of explicit domain restriction (x > 0) for the inequality direction is a logical gap in the setup.
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.