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Limits using Taylor series

Problem 7.670 · medium

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

Near 0, -x + sin(x) = x**5/120 - x**3/6 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{5}}{120} + \frac{x^{3}}{6} - x + \sin{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{\frac{x^{5}}{60} - \frac{x^{3}}{3}}{x^{3}} = \frac{x^{2}}{60} - \frac{1}{3} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{- 2 x + 2 \sin{\left(x \right)}}{x^{3}}\right) = - \frac{1}{3} \]
Let x → 0: only the constant term survives.✓ Proved

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

Lines: 3 proved, 1 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 expansion and applies it to evaluate the limit. The algebraic manipulation and final limit evaluation are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the Maclaurin series expansion and applies it to evaluate the limit. The algebraic manipulation and final limit evaluation are correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series for sin(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Line 1 contains a sign error in the Maclaurin series expansion: it claims -x + sin(x) = x^5/120 - x^3/6, but the correct expansion is -x^3/6 + x^5/120 (the x^3 term is negative). Although the final answer is correct, the intermediate algebraic statement is false.
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.