The squeeze theorem
Problem 1.736 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -2 + \frac{\cos{\left(x \right)}}{x} \).
- Since −1 ≤ cos(·) ≤ 1, the expression lies between -2 - 1/x and -2 + 1/x.
- \[ \lim_{x \to \infty}\left(-2 - \frac{1}{x}\right) = -2 \]The lower bound's limit.✓ Proved
- \[ \lim_{x \to \infty}\left(-2 + \frac{1}{x}\right) = -2 \]The upper bound's limit, the same.✓ Proved
- Both bounds tend to the same value, so the expression trapped between them does too.
Answer \( -2 \)
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: passqwen3.6:27b-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. Thus, the expression is bounded by -2 - 1/x and -2 + 1/x. However, for x > 0, -1/x is negative, so -2 - 1/x is less than -2 + 1/x. The solution claims the expression lies between -2 - 1/x and -2 + 1/x, which is technically correct as an interval, but the phrasing 'lower bound' and 'upper bound' in steps 2 and 3 implies a specific ordering. More critically, the squeeze theorem requires f(x) <= g(x) <= h(x). Here, if we let g(x) = -2 + cos(x)/x, then -2 - 1/x <= g(x) <= -2 + 1/x holds for x > 0. The limits of the bounds are both -2. The logic is actually sound for x -> infinity (where x is positive). Wait, let me re-read carefully. Step 1 says: 'Since -1 <= cos(x) <= 1, the expression lies between -2 - 1/x and -2 + 1/x.' If x > 0, then -1/x <= cos(x)/x <= 1/x. Adding -2: -2 - 1/x <= -2 + cos(x)/x <= -2 + 1/x. So the lower bound is -2 - 1/x and the upper bound is -2 + 1/x. Step 2 calculates limit of -2 - 1/x as -2. Step 3 calculates limit of -2 + 1/x as -2. Step 4 concludes the limit is -2. Is there an error? Let's check the direction. As x -> infinity, x is positive. -1 <= cos(x) <= 1 Divide by x (positive): -1/x <= cos(x)/x <= 1/x. Subtract 2: -2 - 1/x <= -2 + cos(x)/x <= -2 + 1/x. The bounds are correct. The limits are correct. The conclusion is correct. Why did I think it might be wrong? Maybe the 'dir'='- in the equation check? Limit(-2 - 1/x, x, oo, dir='-') Usually 'dir'='-' means approaching from the left. But x -> infinity is a one-sided limit from the right (positive infinity). Approaching infinity 'from the left' doesn't make standard sense in this context unless it means x -> +infinity. In many CAS systems, 'dir'='-' for infinity might be ignored or interpreted as standard limit. However, the sentence says 'Since -1 <= cos(x) <= 1...'. This is true for all real x. If x were negative, the inequalities would flip. But the limit is x -> infinity, which implies x -> +infinity. So the setup is correct for the domain of interest. Let's look closer at Step 1. 'the expression lies between -2 - 1/x and -2 + 1/x'. This is true for x > 0. Is it possible the user considers x -> -infinity? No, problem says x -> infinity. Is there a subtle error? Maybe the bounds are not tight enough? No, squeeze theorem doesn't require tight bounds, just that they converge to the same value. Let's re-read the prompt's instruction: 'find what is wrong'. If the solution is correct, I should say pass. But wait. Step 2: `Limit(-2 - 1/x, x, oo, dir='-') = -2` Step 3: `Limit(-2 + 1/x, x, oo, dir='-') = -2` The `dir='-'` flag is suspicious. For `x -> oo`, the direction is inherently from below (increasing x). Some systems use `dir='-'` to mean `x -> -oo`? Or `x -> c-`? If `dir='-'` means approaching from the left, for `x -> oo`, it's ambiguous. But usually `x -> oo` is a single limit. However, the sentence in Step 1 does not restrict x to be positive. It states a general fact. If a student reads Step 1, they might think it holds for all x. If x < 0, -1/x > 1/x. So -2 - 1/x > -2 + 1/x. The 'lower' bound would be -2 + 1/x and 'upper' would be -2 - 1/x. The sentence 'lies between A and B' is symmetric, so it's still true. But Step 2 calls -2 - 1/x the 'lower bound'. For x < 0, -2 - 1/x is the UPPER bound. Since the limit is x -> infinity, we only care about x > 0. Does the solution explicitly state x > 0? No. It says 'Since -1 <= cos(x) <= 1...'. This implication is only valid for the specific inequality chain if we assume x > 0 to maintain the order of inequalities when dividing. If the student applies this logic to x -> -infinity, they would get the bounds reversed but the same limit. However, strictly speaking, Step 1 implies the inequality -2 - 1/x <= ... <= -2 + 1/x. This inequality is FALSE for x < 0. For x < 0, -1/x > 0 and 1/x < 0. Example x = -1. -2 - (-1) = -1. -2 + (-1) = -3. Expression: -2 + cos(-1)/(-1) = -2 - cos(1) approx -2.54. Is -1 <= -2.54 <= -3? No. So the inequality stated in Step 1 is NOT universally true. It is only true for x > 0. The solution fails to specify that x > 0 (or x is sufficiently large positive) for the inequality to hold in that order. While the limit is at +infinity, a rigorous proof should specify the domain where the squeeze holds. Is this an 'error' or 'misleading'? It's a missing hypothesis for the inequality. If a student copies this, they might think -2 - 1/x is always the lower bound. This is false. Therefore, it is misleading/wrong. Let's check the severity. The conclusion is correct. The method is correct for the limit in question. But the statement 'the expression lies between -2 - 1/x and -2 + 1/x' is presented as a direct consequence of '-1 <= cos(x) <= 1'. This deduction is flawed without the condition x > 0. I will mark it as an error because the logical step from 'cos bounded' to 'expression bounded by these specific functions' is invalid without restricting the domain, and the solution omits this restriction.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-20 — [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. Thus, the expression is bounded by -2 - 1/x and -2 + 1/x. However, for x > 0, -1/x is negative, so -2 - 1/x is less than -2 + 1/x. The solution claims the expression lies between -2 - 1/x and -2 + 1/x, which is technically correct as an interval, but the phrasing 'lower bound' and 'upper bound' in steps 2 and 3 implies a specific ordering. More critically, the squeeze theorem requires f(x) <= g(x) <= h(x). Here, if we let g(x) = -2 + cos(x)/x, then -2 - 1/x <= g(x) <= -2 + 1/x holds for x > 0. The limits of the bounds are both -2. The logic is actually sound for x -> infinity (where x is positive). Wait, let me re-read carefully. Step 1 says: 'Since -1 <= cos(x) <= 1, the expression lies between -2 - 1/x and -2 + 1/x.' If x > 0, then -1/x <= cos(x)/x <= 1/x. Adding -2: -2 - 1/x <= -2 + cos(x)/x <= -2 + 1/x. So the lower bound is -2 - 1/x and the upper bound is -2 + 1/x. Step 2 calculates limit of -2 - 1/x as -2. Step 3 calculates limit of -2 + 1/x as -2. Step 4 concludes the limit is -2. Is there an error? Let's check the direction. As x -> infinity, x is positive. -1 <= cos(x) <= 1 Divide by x (positive): -1/x <= cos(x)/x <= 1/x. Subtract 2: -2 - 1/x <= -2 + cos(x)/x <= -2 + 1/x. The bounds are correct. The limits are correct. The conclusion is correct. Why did I think it might be wrong? Maybe the 'dir'='- in the equation check? Limit(-2 - 1/x, x, oo, dir='-') Usually 'dir'='-' means approaching from the left. But x -> infinity is a one-sided limit from the right (positive infinity). Approaching infinity 'from the left' doesn't make standard sense in this context unless it means x -> +infinity. In many CAS systems, 'dir'='-' for infinity might be ignored or interpreted as standard limit. However, the sentence says 'Since -1 <= cos(x) <= 1...'. This is true for all real x. If x were negative, the inequalities would flip. But the limit is x -> infinity, which implies x -> +infinity. So the setup is correct for the domain of interest. Let's look closer at Step 1. 'the expression lies between -2 - 1/x and -2 + 1/x'. This is true for x > 0. Is it possible the user considers x -> -infinity? No, problem says x -> infinity. Is there a subtle error? Maybe the bounds are not tight enough? No, squeeze theorem doesn't require tight bounds, just that they converge to the same value. Let's re-read the prompt's instruction: 'find what is wrong'. If the solution is correct, I should say pass. But wait. Step 2: `Limit(-2 - 1/x, x, oo, dir='-') = -2` Step 3: `Limit(-2 + 1/x, x, oo, dir='-') = -2` The `dir='-'` flag is suspicious. For `x -> oo`, the direction is inherently from below (increasing x). Some systems use `dir='-'` to mean `x -> -oo`? Or `x -> c-`? If `dir='-'` means approaching from the left, for `x -> oo`, it's ambiguous. But usually `x -> oo` is a single limit. However, the sentence in Step 1 does not restrict x to be positive. It states a general fact. If a student reads Step 1, they might think it holds for all x. If x < 0, -1/x > 1/x. So -2 - 1/x > -2 + 1/x. The 'lower' bound would be -2 + 1/x and 'upper' would be -2 - 1/x. The sentence 'lies between A and B' is symmetric, so it's still true. But Step 2 calls -2 - 1/x the 'lower bound'. For x < 0, -2 - 1/x is the UPPER bound. Since the limit is x -> infinity, we only care about x > 0. Does the solution explicitly state x > 0? No. It says 'Since -1 <= cos(x) <= 1...'. This implication is only valid for the specific inequality chain if we assume x > 0 to maintain the order of inequalities when dividing. If the student applies this logic to x -> -infinity, they would get the bounds reversed but the same limit. However, strictly speaking, Step 1 implies the inequality -2 - 1/x <= ... <= -2 + 1/x. This inequality is FALSE for x < 0. For x < 0, -1/x > 0 and 1/x < 0. Example x = -1. -2 - (-1) = -1. -2 + (-1) = -3. Expression: -2 + cos(-1)/(-1) = -2 - cos(1) approx -2.54. Is -1 <= -2.54 <= -3? No. So the inequality stated in Step 1 is NOT universally true. It is only true for x > 0. The solution fails to specify that x > 0 (or x is sufficiently large positive) for the inequality to hold in that order. While the limit is at +infinity, a rigorous proof should specify the domain where the squeeze holds. Is this an 'error' or 'misleading'? It's a missing hypothesis for the inequality. If a student copies this, they might think -2 - 1/x is always the lower bound. This is false. Therefore, it is misleading/wrong. Let's check the severity. The conclusion is correct. The method is correct for the limit in question. But the statement 'the expression lies between -2 - 1/x and -2 + 1/x' is presented as a direct consequence of '-1 <= cos(x) <= 1'. This deduction is flawed without the condition x > 0. I will mark it as an error because the logical step from 'cos bounded' to 'expression bounded by these specific functions' is invalid without restricting the domain, and the solution omits this restriction.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: pass 2026-10-19gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The inequality in step 1 is incorrect for x > 0. Since 1/x > 0, multiplying -1 <= cos(x) <= 1 by 1/x gives -1/x <= cos(x)/x <= 1/x. Adding -2 yields -2 - 1/x <= -2 + cos(x)/x <= -2 + 1/x. However, the solution claims the expression lies between -2 - 1/x and -2 + 1/x, which is technically correct as a range, but the phrasing 'lies between' usually implies the specific bounds derived. More critically, the limits in steps 2 and 3 are computed as x -> oo, but the syntax `dir='-'` suggests a one-sided limit from the left, which is irrelevant for x -> oo. The main issue is that the squeeze theorem application is valid, but the description in step 1 is slightly imprecise regarding the direction of the inequality if x were negative, though here x->oo. Wait, let's look closer. -1 <= cos(x) <= 1. Divide by x (positive): -1/x <= cos(x)/x <= 1/x. Add -2: -2 - 1/x <= -2 + cos(x)/x <= -2 + 1/x. The solution says 'lies between -2 - 1/x and -2 + 1/x'. This is correct. The limits are both -2. The conclusion is correct. Why would it fail? Ah, look at step 2 and 3. `Limit(..., x, oo, dir='-')`. The direction `dir='-'` is nonsensical for infinity. It should be `dir='+'` or omitted. But more importantly, is there a subtle error? No, the math is sound. Let me re-read the prompt constraints. 'Check that... each theorem or test is applied where its hypotheses hold'. The squeeze theorem requires f(x) <= g(x) <= h(x). The solution states the bounds. The limits match. The conclusion follows. Is it 'pass'? Let's check the bounds again. Lower bound: -2 - 1/x. Upper bound: -2 + 1/x. As x->oo, 1/x -> 0. Limits are -2. Correct. Why did I think it might fail? Maybe the 'dir' parameter is a syntax error in the 'equation' line which I am told not to re-check algebra of, but I should check if the sentence setup is wrong. The sentence says 'Since -1 <= cos <= 1, the expression lies between...'. This is a valid deduction. The equations are marked 'unchecked' but the prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The equations are marked 'unchecked' in the text provided by the user, but the instructions say 'do not re-check the algebra of a line marked proved'. The user's text says '[equation, unchecked]'. This is ambiguous. Usually, 'unchecked' means the CAS hasn't verified it, or it's a placeholder. But the instruction says 'do not re-check... a line marked proved'. These are marked 'unchecked'. Does that mean I should check them? Or does it mean they are not 'proved' lines? The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. It doesn't explicitly say I must check equations. It says 'The sentences are what you are here for.' So I should focus on the sentences. Sentence 1: Correct. Sentence 4: Correct. So it should be pass? Let me look for a trap. 'dir='-''. If the CAS checked it, it might have interpreted it as x->-oo? No, x->oo. If the limit was taken as x->-oo, 1/x -> 0 from below. -2 - 1/x -> -2. -2 + 1/x -> -2. The limits are still -2. So even if the direction was wrong, the limit value is correct. Is there a sign error? -2 + cos(x)/x. Bounds: -2 - 1/x and -2 + 1/x. Correct. I will mark it pass.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.