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Limits using Taylor series

Problem 7.603 · medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{1 - \cos{\left(x \right)}}{x^{2}} \).

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

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

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	✓ 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: pass — The solution correctly identifies the Maclaurin series for 1 - cos(x) and applies the limit properly. The intermediate step involving the error term is valid and rigorous.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the Maclaurin series for 1 - cos(x) and applies the limit properly. The intermediate step involving the error term is valid and rigorous.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the Maclaurin series for 1 - cos(x) and uses it to evaluate the limit. Although Step 1 writes the series for cos(x) - 1, the subsequent steps correctly handle the sign to arrive at the limit for (1 - cos(x))/x^2. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the Maclaurin series for cosine, performs the algebraic simplification, and takes the limit. The intermediate step involving the limit of the remainder term (line 2) is technically correct in spirit (showing the error is higher order), though the notation is slightly non-standard; however, it does not lead to a false conclusion.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.