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The squeeze theorem

Problem 1.704 · medium

Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \cos{\left(\frac{4}{x} \right)} \).

Since −1 ≤ cos(·) ≤ 1, the expression lies between -x**2*Abs(x) and x**2*Abs(x).
\[ \lim_{x \to 0}\left(- x^{2} \left|{x}\right|\right) = 0 \]
The lower bound's limit.✓ Proved
\[ \lim_{x \to 0}\left(x^{2} \left|{x}\right|\right) = 0 \]
The upper bound's limit, the same.✓ Proved
Both bounds tend to the same value, so the expression trapped between them does too.

Answer \( 0 \)

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 (error) — The bounds derived in step 1 are incorrect; multiplying -1 <= cos(4/x) <= 1 by x^3 yields -|x|^3 <= x^3 cos(4/x) <= |x|^3, not -x^2|x|. While the final limit is correct, the squeeze theorem is applied to the wrong functions.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The bounds derived in step 1 are incorrect; multiplying -1 <= cos(4/x) <= 1 by x^3 yields -|x|^3 <= x^3 cos(4/x) <= |x|^3, not -x^2|x|. While the final limit is correct, the squeeze theorem is applied to the wrong functions.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Squeeze Theorem by bounding the cosine term and showing the resulting bounds converge to 0.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The bounds derived in step 1 are incorrect. Since -1 <= cos(4/x) <= 1, multiplying by x^3 yields -|x^3| <= x^3 cos(4/x) <= |x^3|, not +/- x^2 |x|. Although the limit of the incorrect bounds is also 0, the squeeze theorem requires the function to be actually bounded by the functions used, which is not true here (e.g., at x=0.1, x^3 cos(4/x) can be 0.001, but the upper bound x^2|x| is 0.001, wait, actually x^3 = x^2 * |x| for x>0? No. x^3 = x^2 * x. If x is negative, x^3 is negative. Let's check x=0.1. x^3 = 0.001. x^2|x| = 0.01. So 0.001 <= 0.01. This holds. Let's check x=-0.1. x^3 = -0.001. Lower bound -x^2|x| = -0.01. -0.01 <= -0.001. This holds. So the bounds are valid? Wait. The statement says the expression lies between -x^2|Abs(x)| and x^2|Abs(x)|. Is |x^3| <= x^2|x|? |x^3| = |x|^3. x^2|x| = |x|^3. They are equal. So the bounds are actually correct? Let me re-read carefully. Step 1 says: 'Since -1 <= cos <= 1, the expression lies between -x^2*Abs(x) and x^2*Abs(x)'. The expression is x^3 cos(4/x). The absolute value is |x^3 cos(4/x)| <= |x^3| = |x|^3. The bounds given are +/- |x|^3. So the bounds are correct. Why did I think they were wrong? Ah, I misread x^2*Abs(x) as something different from |x|^3. x^2 is always positive, Abs(x) is positive. x^2*Abs(x) = |x|^2 * |x| = |x|^3. So the bounds are correct. The limit is 0. The logic is sound. Wait, is there a subtlety? 'dir=+-' usually means two-sided. The limits are 0. The squeeze theorem applies. Is there any error? Let's look closer. Step 1: 'Since -1 <= cos(·) <= 1, the expression lies between -x**2*Abs(x) and x**2*Abs(x).' This implies -|x|^3 <= x^3 cos(4/x) <= |x|^3. This is true. Step 2 and 3 calculate limits of +/- |x|^3 as x->0, which are 0. Step 4 concludes the limit is 0. This seems correct. Why would it fail? Maybe the notation 'dir=+-' is non-standard or misleading? Or maybe the prompt implies I should find a subtle error. Let's re-evaluate. Is it possible that for some x, x^3 cos(4/x) exceeds |x|^3? No, because |cos| <= 1. So |x^3 cos| <= |x^3|. The bounds are tight. The solution is correct. I will mark it as 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 2026-10-19 with SymPy 1.14.0.