The squeeze theorem
Problem 1.812 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 3 + \frac{\sin{\left(4 x \right)}}{x} \).
- Since −1 ≤ sin(·) ≤ 1, the expression lies between 3 - 1/x and 3 + 1/x.Reviewed
- \[ \lim_{x \to \infty}\left(3 - \frac{1}{x}\right) = 3 \]The lower bound's limit.✓ Proved
- \[ \lim_{x \to \infty}\left(3 + \frac{1}{x}\right) = 3 \]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 \( 3 \)
✓ 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 sine term and showing that both bounds converge to 3.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the sine term and showing that both bounds converge to 3.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: fail (style) 2026-11-07 — [domain objection, downgraded to style] The squeeze theorem requires the bounds to hold for all x in the neighborhood of the limit. Since x approaches infinity, x is positive, so 1/x > 0. However, the solution fails to explicitly divide sin(4x) by x to establish the inequality 3 - 1/x <= 3 + sin(4x)/x <= 3 + 1/x. While the resulting bounds are correct for x > 0, the sentence claims the expression 'lies between' these bounds based solely on -1 <= sin <= 1, omitting the crucial division step and the sign of x, which is a logical gap in the proof structure.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: fail (style) 2026-11-07 — [domain objection, downgraded to style] The squeeze theorem requires the inequality to hold for all x in a neighborhood of the limit. For x < 0, dividing by x reverses the inequality, so the bounds are swapped. The solution fails to restrict x > 0, making the setup technically incorrect for the general case, though the limit is correct.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.