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The squeeze theorem

Problem 1.707 · medium

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

Since −1 ≤ cos(·) ≤ 1, the expression lies between 3 - 1/x and 3 + 1/x.
Reviewed
\[ \lim_{x \to \infty}\left(3 - \frac{1}{x}\right) = 3 \]
The lower bound's limit.✓ Proved
\[ \lim_{x \to \infty}\left(3 + \frac{1}{x}\right) = 3 \]
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 \( 3 \)

✓ 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 by establishing valid bounds for the cosine term and showing that the limits of the bounding functions converge to the same value.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the squeeze theorem by establishing valid bounds for the cosine term and showing that the limits of the bounding functions converge to the same value.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution fails to restrict the inequality to x > 0. For x < 0, dividing by x reverses the inequalities, so the expression is not bounded by 3 - 1/x and 3 + 1/x in that region. Although the limit is at infinity (where x is positive), the statement 'the expression lies between...' is technically false without specifying the domain.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The inequality in step 1 is invalid for x approaching positive infinity because dividing by a negative x reverses the inequalities. Specifically, for x < 0, 1/x is negative, so -1/x > 1/x, meaning the 'lower' bound is actually greater than the 'upper' bound.
gpt-oss:20b: pass 2026-10-15

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 2026-10-19 with SymPy 1.14.0.