Home›Calculus 1›The squeeze theorem›Problem 1.705

The squeeze theorem

Problem 1.705 · medium

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

Since −1 ≤ cos(·) ≤ 1, the expression lies between -3 - 1/x and -3 + 1/x.
\[ \lim_{x \to \infty}\left(-3 - \frac{1}{x}\right) = -3 \]
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.

Answer \( -3 \)

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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The lower bound is incorrect for x > 0. Since -1 <= cos(x) <= 1, dividing by positive x gives -1/x <= cos(x)/x <= 1/x. Adding -3 yields -3 - 1/x <= -3 + cos(x)/x <= -3 + 1/x. However, the solution claims the expression lies between -3 - 1/x and -3 + 1/x, which is technically correct as an interval, but the phrasing 'lies between' usually implies the specific bounds derived from the inequality. More critically, the squeeze theorem requires the function to be *sandwiched*. The solution states the expression lies between the bounds, which is true. But wait, let's look closer. -1 <= cos(x) <= 1. For x > 0, -1/x <= cos(x)/x <= 1/x. So -3 - 1/x <= -3 + cos(x)/x <= -3 + 1/x. The solution says 'lies between -3 - 1/x and -3 + 1/x'. This is correct. Why did I think it was wrong? Ah, the direction of the inequality for the lower bound. If x is positive, -1/x is negative. -3 - 1/x is indeed the lower bound. The logic holds. Is there a subtle error? The problem asks for limit as x -> infinity. The bounds are correct. The limits are correct. The conclusion is correct. Why would this fail? Let me re-read carefully. 'Since -1 <= cos(.) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This is a valid deduction for x > 0. The limits are calculated correctly. The squeeze theorem application is correct. Is it possible the issue is 'dir='-'? No, x -> infinity usually implies positive infinity. The limits are -3. The answer is -3. It seems correct. Wait, is there a sign error in the bounds? Lower bound: -3 + (-1)/x = -3 - 1/x. Upper bound: -3 + 1/x. Yes. Okay, let's look for a 'misleading' aspect. The sentence 'the expression lies between...' is slightly informal but standard. Is there a case where x is negative? The limit is x -> infinity, so we only care about large positive x. The setup is sound. Let me check the JSON format requirements again. If it's sound, pass. But I must be adversarial. Is there any ambiguity? 'dir='-' in the equation comments suggests approaching from the left, but x -> oo is from the right. However, the equation is marked 'proved' by a CAS, so I shouldn't check the algebra or the CAS input parameters unless they contradict the math. The math result -3 is correct. The sentence logic is correct. I will pass it.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The lower bound is incorrect for x > 0. Since -1 <= cos(x) <= 1, dividing by positive x gives -1/x <= cos(x)/x <= 1/x. Adding -3 yields -3 - 1/x <= -3 + cos(x)/x <= -3 + 1/x. However, the solution claims the expression lies between -3 - 1/x and -3 + 1/x, which is technically correct as an interval, but the phrasing 'lies between' usually implies the specific bounds derived from the inequality. More critically, the squeeze theorem requires the function to be *sandwiched*. The solution states the expression lies between the bounds, which is true. But wait, let's look closer. -1 <= cos(x) <= 1. For x > 0, -1/x <= cos(x)/x <= 1/x. So -3 - 1/x <= -3 + cos(x)/x <= -3 + 1/x. The solution says 'lies between -3 - 1/x and -3 + 1/x'. This is correct. Why did I think it was wrong? Ah, the direction of the inequality for the lower bound. If x is positive, -1/x is negative. -3 - 1/x is indeed the lower bound. The logic holds. Is there a subtle error? The problem asks for limit as x -> infinity. The bounds are correct. The limits are correct. The conclusion is correct. Why would this fail? Let me re-read carefully. 'Since -1 <= cos(.) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This is a valid deduction for x > 0. The limits are calculated correctly. The squeeze theorem application is correct. Is it possible the issue is 'dir='-'? No, x -> infinity usually implies positive infinity. The limits are -3. The answer is -3. It seems correct. Wait, is there a sign error in the bounds? Lower bound: -3 + (-1)/x = -3 - 1/x. Upper bound: -3 + 1/x. Yes. Okay, let's look for a 'misleading' aspect. The sentence 'the expression lies between...' is slightly informal but standard. Is there a case where x is negative? The limit is x -> infinity, so we only care about large positive x. The setup is sound. Let me check the JSON format requirements again. If it's sound, pass. But I must be adversarial. Is there any ambiguity? 'dir='-' in the equation comments suggests approaching from the left, but x -> oo is from the right. However, the equation is marked 'proved' by a CAS, so I shouldn't check the algebra or the CAS input parameters unless they contradict the math. The math result -3 is correct. The sentence logic is correct. I will pass it.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The inequality in step 1 is incorrect for x > 0. Since 1/x > 0, multiplying -1 <= cos(x) <= 1 by 1/x yields -1/x <= cos(x)/x <= 1/x. Adding -3
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the Squeeze Theorem. The bounds are derived correctly from the range of cosine, and the limits of the bounds are correctly identified as -3.
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.