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The squeeze theorem

Problem 1.735 · medium

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

Since −1 ≤ sin(·) ≤ 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: fail (error) — The bounds derived in step 1 are incorrect; x^3 * sin(3/x) is bounded by -|x|^3 and |x|^3, not -x^2| x | and x^2| x |. Although the limits of the incorrect bounds also happen to be 0, the inequality used to trap the function is false for |x| != 1.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-20 — The bounds derived in step 1 are incorrect; x^3 * sin(3/x) is bounded by -|x|^3 and |x|^3, not -x^2| x | and x^2| x |. Although the limits of the incorrect bounds also happen to be 0, the inequality used to trap the function is false for |x| != 1.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The bounds in step 1 are incorrect; multiplying -1 <= sin(3/x) <= 1 by x^3 yields -|x^3| <= x^3 sin(3/x) <= |x^3|, not bounds involving x^2| x |. Although the final limit is correct, the squeeze theorem is applied with the wrong bounding functions.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The bounds derived in step 1 are incorrect. Since |sin(3/x)| <= 1, the expression x^3 sin(3/x) is bounded by -|x^3| and |x^3|, not -x^2|x| and x^2|x|. Although the limits of the incorrect bounds also happen to be 0, the squeeze theorem requires the function to actually be trapped between the chosen bounds, which is false here (e.g., at x=0.1, x^3 sin(3/x) can be larger than x^2|x|).
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.