The squeeze theorem
Problem 1.626 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(2 n \right)}}{n} \).
- Since −1 ≤ cos(·) ≤ 1, the expression lies between -1/n and 1/n.Reviewed
- \[ \lim_{n \to \infty}\left(- \frac{1}{n}\right) = 0 \]The lower bound's limit.✓ Proved
- \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]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 for the cosine term and showing that both bounds 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 bounds converge to 0.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Squeeze Theorem with valid bounds and limits.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the cosine term and showing that both 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.