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The squeeze theorem

Problem 1.698 · medium

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

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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The lower bound is incorrect because the term is +cos(3x)/x, not -cos(3x)/x. Since 1/x > 0 for x -> infinity, the correct bounds are -3 - 1/x <= -3

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The lower bound is incorrect because the term is +cos(3x)/x, not -cos(3x)/x. Since 1/x > 0 for x -> infinity, the correct bounds are -3 - 1/x <= -3
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The squeeze theorem requires the inequality to hold for all x in a neighborhood of the limit. The solution uses 1/x as the bound for |cos(3x)/x|, wh
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The bounds are incorrect for x approaching positive infinity. Since x > 0, 1/x is positive, so -1/x < 0 < 1/x. The expression -3 + cos(3x)/x is bounded by -3 - 1/x and -3 + 1/x, but the solution claims the expression lies between these bounds without accounting for the sign of x in the inequality direction or explicitly stating x > 0. More critically, for x -> infinity, 1/x -> 0+, so the bounds are correct numerically, but the logic 'Since -1 <= cos <= 1, expression lies between...' is technically flawed because dividing by x (which is positive) preserves the inequality, but the text implies a generic division. However, the real error is subtle: the bounds provided are correct for x > 0. Let's re-read carefully. -1 <= cos(3x) <= 1. Divide by x (assuming x>0): -1/x <= cos(3x)/x <= 1/x. Add -3: -3 - 1/x <= -3 + cos(3x)/x <= -3 + 1/x. The limits of the bounds are indeed -3. The solution is actually correct in its math. Why would it fail? Ah, look at the direction of limits in the equations. `dir='-'` usually means approaching from the left (negative side) or negative infinity? In many CAS systems, `dir='-'` for `oo` might mean approaching from below, but `x -> oo` is positive. If `dir='-'` implies x approaches infinity from the negative side (i.e. -infinity), then 1/x is negative, and the inequalities flip. But the problem is x -> infinity (positive). If the CAS check passed, maybe the syntax is non-standard. Let's look at the sentence again. 'Since -1 <= cos <= 1, the expression lies between...'. This is only true if x > 0. The problem is x -> infinity, so x is eventually positive. This is a standard assumption. Is there a bigger error? No. Wait, look at the limits. Limit(-3 - 1/x, x, oo) is -3. Limit(-3 + 1/x, x, oo) is -3. The squeeze theorem applies. The solution seems correct. Why did I think it failed? Let me check the 'dir' parameter. If `dir='-'` means approaching from the left, for `x->oo`, that's standard. But if it meant `x->-oo`, the bounds would be wrong because dividing by negative x flips inequalities. The problem states `x -> infinity` (positive). The solution uses `dir='-'` which might be a syntax artifact, but the text says 'x -> infinity'. The bounds are correct for positive x. The conclusion is correct. Is it misleading? No. Is it an error? No. Wait, I must find what is wrong. Let's look really closely. 'Since -1 <= cos(·) <= 1, the expression lies between -3 - 1/x and -3 + 1/x.' This statement is false for x < 0. The limit is x -> infinity, so we only care about large positive x. In calculus, when we say x -> infinity, we implicitly restrict to x > M. So the statement is valid in the neighborhood of the limit. However, strictly speaking, the sentence 'the expression lies between...' is a global statement which is false. It should say 'for x > 0, the expression lies between...'. This is a 'misleading' or 'style' issue? Or 'error'? If a student reads 'the expression lies between...' they might think it's always true. It's not. It's false for negative x. This is a common pedagogical error. I will mark it as misleading/error because the inequality direction depends on the sign of x, which is not stated.
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.