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Limits using Taylor series

Problem 7.704 · medium

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

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
\[ \frac{\frac{x^{5}}{5} - \frac{x^{3}}{3}}{x^{3}} = \frac{x^{2}}{5} - \frac{1}{3} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( - \frac{1}{3} \)

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) — Line 1 states the expansion for x - atan(x), but the problem requires the expansion for -x + atan(x). The solution implicitly flips the sign in subsequent steps without explicitly stating that -x + atan(x) = -(x - atan(x)), leading to a confusing and technically incorrect setup in the first sentence.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — Line 1 states the expansion for x - atan(x), but the problem requires the expansion for -x + atan(x). The solution implicitly flips the sign in subsequent steps without explicitly stating that -x + atan(x) = -(x - atan(x)), leading to a confusing and technically incorrect setup in the first sentence.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series expansion and applies limit laws to find the result. The notation in step 1 is slightly unconventional but mathematically sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly states the Maclaurin series for x - atan(x) as -x^5/5 + x^3/3, omitting the leading term x. The actual expansion is x - (x - x^3/3 + x^5/5 - ...) = x^3/3 - x^5/5 + ..., so the numerator in the limit is x^3/3 - x^5/5 + ..., not -x^5/5 + x^3/3 + ... (which implies a missing x term or incorrect sign/ordering). While the final answer -1/3 is correct, the intermediate series representation is mathematically false and misleading.
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 2026-10-19 with SymPy 1.14.0.