Limits using Taylor series
Problem 7.699 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- x + \sin{\left(x \right)}}{x^{3}} \).
- 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
- \[ \frac{\frac{x^{5}}{120} - \frac{x^{3}}{6}}{x^{3}} = \frac{x^{2}}{120} - \frac{1}{6} \]Divide through by the denominator.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- x + \sin{\left(x \right)}}{x^{3}}\right) = - \frac{1}{6} \]Let x → 0: only the constant term survives.✓ Proved
Answer \( - \frac{1}{6} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the Maclaurin series expansion, performs the algebraic simplification, and evaluates the limit. The intermediate step involving the limit of the remainder term is a valid rigorous justification.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the Maclaurin series expansion, performs the algebraic simplification, and evaluates the limit. The intermediate step involving the limit of the remainder term is a valid rigorous justification.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series expansion and applies limit laws properly to find the result.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the Maclaurin series for sin(x), identifies the leading non-zero term in the numerator, and correctly evaluates the limit.gpt-oss:20b: fail (misleading) 2026-10-15 — Sentence 2 incorrectly constructs the expression (-x**5/120 + x**3/6 - x + sin(x))/x**5, which is not a valid step in the limit computation and may confuse students. The correct approach is to directly use the series -x + sin(x) = -x^3/6 + x^5/120 and divide by x^3.
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.