The squeeze theorem
Problem 1.695 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \cos{\left(\frac{3}{x} \right)} \).
- 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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The bounds derived in step 1 are incorrect. Since $-1 \le \cos(3/x) \le 1$, multiplying by $x^3$ yields $-x^3 \le x^3 \cos(3/x) \le x^3$. The solution incorrectly uses $\pm x^2 |x| = \pm |x|^3$, which, while having the same limit, misrepresents the actual bounds (e.g., for $x<0$, $-x^3$ is positive, but $-x^2|x|$ is negative). This is a fundamental algebraic error in setting up the squeeze theorem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The bounds derived in step 1 are incorrect. Since $-1 \le \cos(3/x) \le 1$, multiplying by $x^3$ yields $-x^3 \le x^3 \cos(3/x) \le x^3$. The solution incorrectly uses $\pm x^2 |x| = \pm |x|^3$, which, while having the same limit, misrepresents the actual bounds (e.g., for $x<0$, $-x^3$ is positive, but $-x^2|x|$ is negative). This is a fundamental algebraic error in setting up the squeeze theorem.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The bounds derived in step 1 are incorrect; multiplying -1 <= cos(3/x) <= 1 by x^3 yields -|x^3| <= x^3 cos(3/x) <= |x^3|, not +/- x^2 |x|. Although the limit of the incorrect bounds is also 0, the squeeze theorem requires the function to actually be trapped between the chosen bounds, which it is not.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The bounds derived in step 1 are incorrect; multiplying -1 <= cos(3/x) <= 1 by x^3 yields -|x|^3 <= x^3 cos(3/x) <= |x|^3, not -x^2|x|. While the final limit is correct, the squeeze theorem is applied with wrong bounding functions.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.