The squeeze theorem
Problem 1.732 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\sin{\left(4 x \right)}}{x} \).
- Since −1 ≤ sin(·) ≤ 1, the expression lies between -1/x and 1/x.
- \[ \lim_{x \to \infty}\left(- \frac{1}{x}\right) = 0 \]The lower bound's limit.✓ Proved
- \[ \lim_{x \to \infty} \frac{1}{x} = 0 \]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: fail (error) — The solution incorrectly uses dir='-' for the limits as x→∞; the correct direction is dir='+' (approaching from the right). This mistake misstates the application of the squeeze theorem.qwen3.6:27b-mlx: pass — The solution correctly applies the Squeeze Theorem. The bounds are valid for x > 0, and the limits of the bounding functions are correctly identified as 0.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies the Squeeze Theorem. The bounds are valid for x > 0, and the limits of the bounding functions are correctly identified as 0.gpt-oss:20b: fail (error) 2026-10-20 — The solution incorrectly uses dir='-' for the limits as x→∞; the correct direction is dir='+' (approaching from the right). This mistake misstates the application of the squeeze theorem.qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the sine function and showing that both bounds converge to 0 as x approaches infinity.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the Squeeze Theorem by establishing valid bounds for the sine function and showing that the limits of these bounds are equal.gpt-oss:20b: pass 2026-10-19
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 2026-10-20 with SymPy 1.14.0.