Home›Calculus 1›The squeeze theorem›Problem 1.738

The squeeze theorem

Problem 1.738 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \cos{\left(\frac{2}{x} \right)} \).

Since −1 ≤ cos(·) ≤ 1, the expression lies between -x**2*Abs(x) and x**2*Abs(x).
\[ \lim_{x \to 0}\left(- x^{2} \left|{x}\right|\right) = 0 \]
The lower bound's limit.✓ Proved
\[ \lim_{x \to 0}\left(x^{2} \left|{x}\right|\right) = 0 \]
The upper bound's limit, the same.✓ Proved
Both bounds tend to the same value, so the expression trapped between them does too.

Answer \( 0 \)

Lines: 2 proved, 2 not checked. 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	Not checked	—	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	Not checked	—	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: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the squeeze theorem by bounding the cosine term and showing the resulting bounds converge to 0.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The bounds derived in step 1 are incorrect; multiplying the inequality -1 <= cos(2/x) <= 1 by x^3 yields -|x|^3 and |x|^3, not -x^2|x| and x^2|x|. Although the limits of the incorrect bounds are also 0, the squeeze theorem requires the function to actually be bounded by the functions whose limits are taken.
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.