The squeeze theorem
Problem 1.818 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \cos{\left(\frac{4}{x} \right)} \).
- Since −1 ≤ cos(·) ≤ 1, the expression lies between -Abs(x) and Abs(x).Reviewed
- \[ \lim_{x \to 0}\left(- \left|{x}\right|\right) = 0 \]The lower bound's limit.✓ Proved
- \[ \lim_{x \to 0} \left|{x}\right| = 0 \]The upper bound's limit, the same.✓ Proved
- Both bounds tend to the same value, so the expression trapped between them does too.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: passqwen3.6:27b-mlx: pass — The solution correctly applies the Squeeze Theorem by establishing valid bounds using the range of the cosine function and showing that both bounds converge to 0.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies the Squeeze Theorem by establishing valid bounds using the range of the cosine function and showing that both bounds converge to 0.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly applies the squeeze theorem by bounding the cosine term and showing the outer terms converge to 0.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly applies the squeeze theorem by bounding the cosine term and showing the outer terms approach zero.gpt-oss:20b: pass 2026-11-07
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 2026-11-08 with SymPy 1.14.0.