The squeeze theorem
Problem 1.709 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 1 + \frac{\cos{\left(x \right)}}{x} \).
- Since −1 ≤ cos(·) ≤ 1, the expression lies between 1 - 1/x and 1 + 1/x.
- \[ \lim_{x \to \infty}\left(1 - \frac{1}{x}\right) = 1 \]The lower bound's limit.✓ Proved
- \[ \lim_{x \to \infty}\left(1 + \frac{1}{x}\right) = 1 \]The upper bound's limit, the same.✓ Proved
- Both bounds tend to the same value, so the expression trapped between them does too.
Answer \( 1 \)
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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to restrict the domain to x > 0. For x < 0, the inequality 1 - 1/x <= 1 + cos(x)/x <= 1 + 1/x is reversed or invalid, so the squeeze theorem cannot be applied directly without specifying that x approaches infinity (implying x > 0) and adjusting the bounds accordingly.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The solution fails to restrict the domain to x > 0. For x < 0, the inequality 1 - 1/x <= 1 + cos(x)/x <= 1 + 1/x is reversed or invalid, so the squeeze theorem cannot be applied directly without specifying that x approaches infinity (implying x > 0) and adjusting the bounds accordingly.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The inequality 1 - 1/x <= 1 + cos(x)/x <= 1 + 1/x is only valid for x > 0. The solution fails to restrict the domain to positive x, which is necessary for the squeeze theorem application as x approaches infinity.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly applies the bounds for 1/x. For x approaching infinity, x is positive, so the inequality -1 <= cos(x) <= 1 implies -1/x <= cos(x)/x <= 1/x. The solution swaps the bounds, claiming the expression is between 1 - 1/x and 1 + 1/x, which is mathematically incorrect (1 - 1/x is actually the upper bound for negative x, but here x->oo). While the final limit is correct, the intermediate inequality setup is wrong.gpt-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.