Home›Calculus 1›The squeeze theorem›Problem 1.701

The squeeze theorem

Problem 1.701 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\cos{\left(2 x \right)}}{x} \).

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 \( -3 \)

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 cosine term and showing that both bounds converge to -3.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the squeeze theorem by establishing valid bounds for the cosine term and showing that both bounds converge to -3.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The squeeze theorem is only valid when the inequality holds in a neighborhood of the limit point. For x < 0, 1/x is negative, so the derived bounds -3 - 1/x and -3 + 1/x are swapped relative to the actual expression, violating the squeeze condition. The solution fails to restrict x to positive values.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The squeeze theorem requires non-negative bounds for the oscillating term when x approaches infinity, but the solution uses -1/x which is negative.
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 2026-10-18 with SymPy 1.14.0.