Limits using Taylor series
Problem 7.698 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 3 x + 3 \sin{\left(x \right)}}{x^{3}} \).
- 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
- \[ \frac{\frac{x^{5}}{40} - \frac{x^{3}}{2}}{x^{3}} = \frac{x^{2}}{40} - \frac{1}{2} \]Divide through by the denominator.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 3 x + 3 \sin{\left(x \right)}}{x^{3}}\right) = - \frac{1}{2} \]Let x → 0: only the constant term survives.✓ Proved
Answer \( - \frac{1}{2} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 order of the remainder is technically unnecessary for the final result but does not introduce any errors.
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 order of the remainder is technically unnecessary for the final result but does not introduce any errors.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series expansion, handles the algebraic simplification and limit evaluation accurately, and arrives at the correct result.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — 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 2026-10-15
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.