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Limits using Taylor series

Problem 7.702 · medium

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

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

Answer \( 1 \)

Lines: 3 proved, 1 reviewed. 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	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 uses it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the Maclaurin series for atan(x) and uses it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series for atan(x) and uses it to evaluate the limit. The algebraic steps and limit evaluation are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The Maclaurin series for atan(x) is x - x^3/3 + x^5/5 - ..., so x - atan(x) = x^3/3 - x^5/5 + ... The solution incorrectly states the series as -x^5/5 + x^3/3 + ..., implying a negative leading term or confusing the sign of the x^3 term in the expansion of atan(x). Although the final numerical answer is correct, the stated series expansion is formally incorrect (sign error in the description of the series terms relative to standard ordering or just sloppy notation that contradicts the standard expansion).
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.