The squeeze theorem
Problem 1.706 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -2 + \frac{\cos{\left(4 x \right)}}{x} \).
- Since −1 ≤ cos(·) ≤ 1, the expression lies between -2 - 1/x and -2 + 1/x.Reviewed
- \[ \lim_{x \to \infty}\left(-2 - \frac{1}{x}\right) = -2 \]The lower bound's limit.✓ Proved
- \[ \lim_{x \to \infty}\left(-2 + \frac{1}{x}\right) = -2 \]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 \( -2 \)
Lines: 2 proved, 2 reviewed. 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 | 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 the limits of the bounding functions converge to the same value.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the cosine term and showing that the limits of the bounding functions converge to the same value.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the squeeze theorem by bounding the cosine term and showing the limits of the bounds converge to -2.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The squeeze theorem requires the bounds to hold in a neighborhood of the limit point. The solution uses 1/x as the bound for |cos(4x)/x|, which is ogpt-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 2026-10-19 with SymPy 1.14.0.