Limits using Taylor series
Problem 7.682 · 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
- \[ \frac{- \frac{x^{5}}{120} + \frac{x^{3}}{6}}{x^{3}} = \frac{1}{6} - \frac{x^{2}}{120} \]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 for sin(x) and applies it to evaluate the limit. The intermediate steps are algebraically sound and logically follow from the series expansion.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the Maclaurin series for sin(x) and applies it to evaluate the limit. The intermediate steps are algebraically sound and logically follow from the series expansion.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series expansion and applies the limit properties accurately to derive the result.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly states the Maclaurin series for -x + sin(x) as starting with x^5/120, omitting the crucial -x^3/6 term. This error is masked by a sign flip in step 3, but the initial series expansion is factually wrong and would mislead a student about the structure of the sine series.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.