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Limits using Taylor series

Problem 7.675 · medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{1 - \cos{\left(x \right)}}{x \sin{\left(x \right)}} \).

Near 0, 1 - cos(x) = -x**4/24 + x**2/2 + ⋯ (its Maclaurin series).
\[ \lim_{x \to 0}\left(\frac{\frac{x^{4}}{24} - \frac{x^{2}}{2} - \cos{\left(x \right)} + 1}{x^{4}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \lim_{x \to 0}\left(\frac{1 - \cos{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \frac{1}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( \frac{1}{2} \)

Lines: 2 proved, 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the expression at x = ±10⁻⁸ in 50-digit arithmetic

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution claims that 'only the constant term survives' when taking the limit, which is mathematically incorrect reasoning for a 0/0 indeterminate form. The limit is determined by the ratio of the leading non-zero terms (x^2/2 divided by x^2), not by a constant term. Additionally, Step 2 computes a limit for a different expression (divided by x^4) that is irrelevant to the final result.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution claims that 'only the constant term survives' when taking the limit, which is mathematically incorrect reasoning for a 0/0 indeterminate form. The limit is determined by the ratio of the leading non-zero terms (x^2/2 divided by x^2), not by a constant term. Additionally, Step 2 computes a limit for a different expression (divided by x^4) that is irrelevant to the final result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims the limit is 1/2 by stating 'only the constant term survives,' which is mathematically nonsensical for a 0/0 indeterminate form. The correct reasoning requires dividing the leading terms of the numerator (x^2/2) and denominator (x^2) to get 1/2. Additionally, Step 2 checks the error term of the numerator against x^4, which is irrelevant to the limit evaluation shown in Step 3.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution claims that 'only the constant term survives' when taking the limit, which is mathematically incorrect reasoning for a 0/0 indeterminate form; one must divide the leading terms of the numerator and denominator. Additionally, the intermediate limit in step 2 is irrelevant to the problem statement.
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/taylor_limit, checked 2026-10-18 with SymPy 1.14.0.