Home›Calculus 1›The squeeze theorem›Problem 1.620

The squeeze theorem

Problem 1.620 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 1 + \frac{\sin{\left(x \right)}}{x} \).

Since −1 ≤ sin(·) ≤ 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. Reviewers found nothing wrong with the explanation. 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: pass
qwen3.6:27b-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 + sin(x)/x <= 1 + 1/x is reversed, so the bounds used in steps 2 and 3 do not actually sandwich the function for all x approaching infinity (or rather, the logic ignores the sign of x which is crucial for the inequality direction).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution fails to restrict the domain to x > 0. For x < 0, the inequality 1 - 1/x <= 1 + sin(x)/x <= 1 + 1/x is reversed, so the bounds used in steps 2 and 3 do not actually sandwich the function for all x approaching infinity (or rather, the logic ignores the sign of x which is crucial for the inequality direction).
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The inequalities for the squeeze theorem are only valid for x > 0. Since the limit is as x approaches infinity, this condition is met, but the solution fails to explicitly state that x is positive, which is required to ensure 1/x is positive and thus -1/x <= sin(x)/x <= 1/x. Without this, the direction of the inequalities for the bounds is not rigorously justified.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The squeeze theorem requires the inequality to hold for all x in a neighborhood of the limit point. For x -> infinity, this requires x > 0, but the solution fails to specify this domain restriction, making the inequality 1 - 1/x <= f(x) <= 1 + 1/x invalid for negative x.
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.