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Limits using Taylor series

Problem 7.599 · medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 - 3 \cos{\left(x \right)}}{x^{2}} \).

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
\[ \frac{- \frac{x^{4}}{8} + \frac{3 x^{2}}{2}}{x^{2}} = \frac{3}{2} - \frac{x^{2}}{8} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{3 - 3 \cos{\left(x \right)}}{x^{2}}\right) = \frac{3}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( \frac{3}{2} \)

✓ 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 cosine, substitutes it into the expression, and evaluates the limit. The intermediate step involving the error term (line 2) is technically correct but unnecessary for the final result, yet it does not introduce any errors or misleading concepts.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the Maclaurin series for cosine, substitutes it into the expression, and evaluates the limit. The intermediate step involving the error term (line 2) is technically correct but unnecessary for the final result, yet it does not introduce any errors or misleading concepts.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the Maclaurin series for cosine, substitutes it into the limit expression, and simplifies to find the limit. The intermediate step involving the remainder term is technically correct but unnecessary for the final result, yet it does not introduce any errors or misleading concepts.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the Maclaurin series for cosine, substitutes it into the limit expression, and simplifies to find the limit. The logic is sound and the steps are clear.
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.