Home›Calculus 1›The squeeze theorem›Problem 1.696

The squeeze theorem

Problem 1.696 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -1 + \frac{\cos{\left(3 x \right)}}{x} \).

Since −1 ≤ cos(·) ≤ 1, the expression lies between -1 - 1/x and -1 + 1/x.
Reviewed
\[ \lim_{x \to \infty}\left(-1 - \frac{1}{x}\right) = -1 \]
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.
Reviewed

Answer \( -1 \)

✓ 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Squeeze Theorem. The bounds are derived correctly from the range of cosine, and the limits of the bounding functions are correctly identified as -1.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The lower bound is incorrect for x > 0. Since -1 <= cos(3x) <= 1, dividing by positive x gives -1/x <= cos(3x)/x <= 1/x. Adding -1 yields -1 - 1/x <= expression <= -1 + 1/x. Wait, the solution says 'lies between -1 - 1/x and -1 + 1/x'. Let's re-read carefully. The solution claims the expression lies between -1 - 1/x and -1 + 1/x. This is actually correct algebraically: -1 + (-1/x) <= -1 + cos(3x)/x <= -1 + (1/x). So the bounds are correct. Why did I think it was wrong? Ah, I misread the sign in my head. Let's look closer. Is there a subtle error? The problem asks for limit as x -> infinity. The bounds are -1 - 1/x and -1 + 1/x. As x -> infinity, 1/x -> 0. So both bounds go to -1. The logic holds. Is there a direction issue? 'dir='-' in the equation check suggests approaching from the left, but x -> infinity usually implies positive infinity. If x -> -infinity, 1/x is negative, so the inequalities flip. However, standard notation x -> infinity implies +infinity. The sentence says 'Since -1 <= cos <= 1, the expression lies between...'. This is true for x > 0. For x < 0, the inequalities reverse. The limit is at infinity, so we only care about large positive x. The setup is valid for the domain of interest. Wait, looking at the provided solution text again: 'Limit(..., dir='-')'. This is a computer algebra system output. The sentence says 'Both bounds tend to the same value'. This is correct. Is there any other error? The prompt asks to find what is wrong. Let's look at the bounds again. Lower bound: -1 - 1/x. Upper bound: -1 + 1/x. For large x, -1 - 1/x is slightly less than -1, and -1 + 1/x is slightly more than -1. The expression is trapped. The limits are both -1. The squeeze theorem applies. It seems correct. Why would it fail? Maybe the 'dir='-' in the equation line is misleading or technically wrong for x->oo? Usually x->oo doesn't have a direction like left/right, it's just infinity. But the sentence doesn't mention direction. The sentence is 'Both bounds tend to the same value...'. This is a correct application of the Squeeze Theorem. Let me double check the 'error' possibility. Is it possible the user considers the bounds wrong? No. Is it possible the user considers the limit wrong? No. Is it possible the user considers the theorem application wrong? No. Let me look for 'misleading'. The solution doesn't explicitly state that x must be positive for the inequalities to hold as written. If a student applies this to x -> -infinity, they would get the bounds reversed and might be confused if they don't adjust. However, the problem is x -> infinity. Standard convention is +infinity. I will mark it as pass unless I see a blatant error. Wait, look at line 1: 'Since -1 <= cos(·) <= 1, the expression lies between -1 - 1/x and -1 + 1/x.' This implication is only true if x > 0. If x is negative, dividing by x reverses the inequality. The solution does not specify x > 0. While the limit is at +infinity, rigorous proofs should specify the domain where the inequality holds. Is this an 'error' or 'style'? It's a minor omission. But wait, look at the equation lines. `Limit(..., dir='-')`. This is weird syntax for infinity. Usually `dir='-'` means approaching from the left (e.g., x -> 0-). For x -> oo, direction is irrelevant or standard. If the CAS output is interpreted as 'approaching infinity from the left' (which doesn't make sense for real line infinity, but maybe in extended reals?), it's just a label. The sentence is the focus. The sentence is logically sound for the context of x->oo. I will give it a pass.
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 2026-10-18 with SymPy 1.14.0.