Home›Calculus 1›The squeeze theorem›Problem 1.627

The squeeze theorem

Problem 1.627 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \cos{\left(\frac{2}{x} \right)} \).

Since −1 ≤ cos(·) ≤ 1, the expression lies between -x**2 and x**2.
Reviewed
\[ \lim_{x \to 0}\left(- x^{2}\right) = 0 \]
The lower bound's limit.✓ Proved
\[ \lim_{x \to 0} x^{2} = 0 \]
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 by establishing valid bounds for the cosine term and showing that both bounding functions converge to 0.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the cosine term and showing that both bounding functions converge to 0.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the cosine term and showing that the bounding functions converge to the same limit.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the cosine term and showing that the bounding functions converge to 0.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.