The squeeze theorem
Problem 1.737 · medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\sin{\left(4 x \right)}}{x} \).
- Since −1 ≤ sin(·) ≤ 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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The bounds in step 1 are incorrect. Since the term is sin(4x)/x and x approaches positive infinity, 1/x is positive, so the correct bounds are -3 - 1/x <= -3 + sin(4x)/x <= -3 + 1/x. However, the solution states the expression lies between -3 - 1/x and -3 + 1/x, which is technically correct for the bounds, but the logic relies on |sin(4x)/x| <= 1/x. The main issue is that step 1 claims the expression lies between those bounds based on -1 <= sin <= 1, but it fails to explicitly divide by x or handle the sign of x properly in the text, though the bounds listed are correct for x > 0. Wait, let's look closer. Step 1 says 'Since -1 <= sin <= 1, the expression lies between -3 - 1/x and -3 + 1/x'. This implies -1/x <= sin(4x)/x <= 1/x. This is only true if x > 0. Since x -> infinity, x is positive, so this is valid. However, usually one writes |sin(4x)/x| <= 1/|x|. The bounds provided are correct. Is there an error? Let's re-read carefully. 'Since -1 <= sin(4x) <= 1'. If x is positive, dividing by x preserves inequalities: -1/x <= sin(4x)/x <= 1/x. Adding -3: -3 - 1/x <= -3 + sin(4x)/x <= -3 + 1/x. The bounds in step 1 are correct. The limits in 2 and 3 are correct. The conclusion in 4 is correct. Why would this fail? Ah, look at the direction of rotation or bounds? No. Is it misleading? The step 1 sentence is slightly loose but mathematically sound for x->oo. Let's look for a subtle error. Maybe the 'dir='-' in the equation comments? The problem is x -> infinity (positive). The equations use dir='-' which usually means approaching from the left, but for infinity, it's irrelevant or standard notation for 'from below' which doesn't apply to infinity in the same way. But the equations are marked 'proved'. The sentences are the target. Step 1: 'Since -1 <= sin(4x) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This is a valid deduction for x > 0. Step 4: 'Both bounds tend to the same value...'. This is the Squeeze Theorem. It seems correct. Is there a 'style' issue? Or is it actually correct? Let's check the bounds again. If x is very large positive, 1/x is small positive. Lower bound -3 - small, Upper bound -3 + small. Correct. What if the user meant x -> -infinity? No, problem says x -> infinity. Okay, is there a hidden error? 'Since -1 <= sin(4x) <= 1'. This is always true. 'the expression lies between...'. This requires dividing by x. The sentence skips the division step, implying it's trivial. This is acceptable. However, often 'error' is reserved for wrong math. Is it wrong? No. Is it misleading? No. Is it style? Maybe. But wait, look at the limits. Limit(-3 - 1/x, x, oo) = -3. Correct. Limit(-3 + 1/x, x, oo) = -3. Correct. Conclusion: -3. Correct. Why would I fail this? Let me look really closely at Step 1. 'Since -1 <= sin(4x) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This statement is only true if x > 0. The limit is x -> infinity, so x is eventually positive. The statement is implicitly assuming x > 0. This is standard. I will mark it pass unless I find a specific flaw. Wait, looking at similar adversarial tasks, sometimes the bounds are flipped or the absolute value is ignored when x could be negative. Here x->oo, so x is positive. It seems correct. Let me double check the 'dir='-' in the equation comments. The prompt says 'do not re-check the algebra of a line marked proved'. So I ignore the equation content. I only check sentences. Sentence 1: Setup. Sentence 4: Conclusion. Both seem correct. I will mark it pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-20 — [domain objection, downgraded to style] The bounds in step 1 are incorrect. Since the term is sin(4x)/x and x approaches positive infinity, 1/x is positive, so the correct bounds are -3 - 1/x <= -3 + sin(4x)/x <= -3 + 1/x. However, the solution states the expression lies between -3 - 1/x and -3 + 1/x, which is technically correct for the bounds, but the logic relies on |sin(4x)/x| <= 1/x. The main issue is that step 1 claims the expression lies between those bounds based on -1 <= sin <= 1, but it fails to explicitly divide by x or handle the sign of x properly in the text, though the bounds listed are correct for x > 0. Wait, let's look closer. Step 1 says 'Since -1 <= sin <= 1, the expression lies between -3 - 1/x and -3 + 1/x'. This implies -1/x <= sin(4x)/x <= 1/x. This is only true if x > 0. Since x -> infinity, x is positive, so this is valid. However, usually one writes |sin(4x)/x| <= 1/|x|. The bounds provided are correct. Is there an error? Let's re-read carefully. 'Since -1 <= sin(4x) <= 1'. If x is positive, dividing by x preserves inequalities: -1/x <= sin(4x)/x <= 1/x. Adding -3: -3 - 1/x <= -3 + sin(4x)/x <= -3 + 1/x. The bounds in step 1 are correct. The limits in 2 and 3 are correct. The conclusion in 4 is correct. Why would this fail? Ah, look at the direction of rotation or bounds? No. Is it misleading? The step 1 sentence is slightly loose but mathematically sound for x->oo. Let's look for a subtle error. Maybe the 'dir='-' in the equation comments? The problem is x -> infinity (positive). The equations use dir='-' which usually means approaching from the left, but for infinity, it's irrelevant or standard notation for 'from below' which doesn't apply to infinity in the same way. But the equations are marked 'proved'. The sentences are the target. Step 1: 'Since -1 <= sin(4x) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This is a valid deduction for x > 0. Step 4: 'Both bounds tend to the same value...'. This is the Squeeze Theorem. It seems correct. Is there a 'style' issue? Or is it actually correct? Let's check the bounds again. If x is very large positive, 1/x is small positive. Lower bound -3 - small, Upper bound -3 + small. Correct. What if the user meant x -> -infinity? No, problem says x -> infinity. Okay, is there a hidden error? 'Since -1 <= sin(4x) <= 1'. This is always true. 'the expression lies between...'. This requires dividing by x. The sentence skips the division step, implying it's trivial. This is acceptable. However, often 'error' is reserved for wrong math. Is it wrong? No. Is it misleading? No. Is it style? Maybe. But wait, look at the limits. Limit(-3 - 1/x, x, oo) = -3. Correct. Limit(-3 + 1/x, x, oo) = -3. Correct. Conclusion: -3. Correct. Why would I fail this? Let me look really closely at Step 1. 'Since -1 <= sin(4x) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This statement is only true if x > 0. The limit is x -> infinity, so x is eventually positive. The statement is implicitly assuming x > 0. This is standard. I will mark it pass unless I find a specific flaw. Wait, looking at similar adversarial tasks, sometimes the bounds are flipped or the absolute value is ignored when x could be negative. Here x->oo, so x is positive. It seems correct. Let me double check the 'dir='-' in the equation comments. The prompt says 'do not re-check the algebra of a line marked proved'. So I ignore the equation content. I only check sentences. Sentence 1: Setup. Sentence 4: Conclusion. Both seem correct. I will mark it pass.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-19 — The bounds derived in step 1 are incorrect because the coefficient 4 inside the sine function is ignored; the correct bounds should be -3 - 4/x and -3 + 4/x. Although the final limit is correct, the specific inequalities stated are false.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.