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Limits using Taylor series

Problem 7.680 · 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).
\[ \lim_{x \to 0}\left(\frac{\frac{x^{5}}{5} - \frac{x^{3}}{3} + x - \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{\frac{2 x^{5}}{5} - \frac{2 x^{3}}{3}}{x^{3}} = \frac{2 x^{2}}{5} - \frac{2}{3} \]
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 \( - \frac{2}{3} \)

Lines: 3 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
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: fail (error) — Sentence 1 states the expansion for x - atan(x) but the numerator is -2x + 2atan(x) = -2(x - atan(x)). The solution fails to explicitly apply this factor of -2 in the text, jumping directly to the simplified form in step 3 without showing the substitution. While the algebra in step 3 is marked proved, the logical bridge in the sentences is missing and potentially confusing.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — Sentence 1 states the expansion for x - atan(x) but the numerator is -2x + 2atan(x) = -2(x - atan(x)). The solution fails to explicitly apply this factor of -2 in the text, jumping directly to the simplified form in step 3 without showing the substitution. While the algebra in step 3 is marked proved, the logical bridge in the sentences is missing and potentially confusing.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series for atan(x), substitutes it into the expression, and evaluates the limit by retaining the lowest order non-zero term. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The Maclaurin series for atan(x) is x - x^3/3 + x^5/5 - ..., so x - atan(x) = x^3/3 - x^5/5 + ... Line 1 incorrectly states the series as -x^5
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.