Home›Calculus 2›Limits using Taylor series›Problem 7.602

Limits using Taylor series

Problem 7.602 · medium

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

Near 0, x - atan(x) = -x**5/5 + x**3/3 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{\frac{x^{5}}{5} - \frac{x^{3}}{3} + x - \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{2 x^{5}}{5} + \frac{2 x^{3}}{3}}{x^{3}} = \frac{2}{3} - \frac{2 x^{2}}{5} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{2 x - 2 \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = \frac{2}{3} \]
Let x → 0: only the constant term survives.✓ Proved

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

✓ 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 atan(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the Maclaurin series for atan(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are sound.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the Maclaurin series for atan(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the Maclaurin series for atan(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.
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.