Limits using Taylor series
Problem 7.700 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 \cos{\left(x \right)} - 3}{x \sin{\left(x \right)}} \).
- Near 0, 1 - cos(x) = -x**4/24 + x**2/2 + ⋯ (its Maclaurin series).
- \[ \lim_{x \to 0}\left(\frac{\frac{x^{4}}{24} - \frac{x^{2}}{2} - \cos{\left(x \right)} + 1}{x^{4}}\right) = 0 \]The polynomial matches the numerator to that order.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{3 \cos{\left(x \right)} - 3}{x \sin{\left(x \right)}}\right) = - \frac{3}{2} \]Let x → 0: only the constant term survives.✓ Proved
Answer \( - \frac{3}{2} \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | 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 |
| 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: fail (misleading) — The solution skips the explicit series expansion of both numerator and denominator and merely states that only the constant term survives, which does not convincingly justify the limit. A student could be misled into thinking the limit follows from a single term without proper justification.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The Maclaurin series for 1 - cos(x) is incorrectly stated as starting with x^2/2 (it should be x^2/2, but the sign in the expression '1 - cos(x) = -
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The Maclaurin series for 1 - cos(x) is incorrectly stated as starting with x^2/2 (it should be x^2/2, but the sign in the expression '1 - cos(x) = -gpt-oss:20b: fail (misleading) 2026-10-19 — The solution skips the explicit series expansion of both numerator and denominator and merely states that only the constant term survives, which does not convincingly justify the limit. A student could be misled into thinking the limit follows from a single term without proper justification.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The Maclaurin series for 1 - cos(x) is incorrectly stated as starting with x^2/2 (it should be x^2/2 - x^4/24 + ...); the signs are flipped. While the final limit is correct, the intermediate series expansion is mathematically false.gpt-oss:20b: fail (misleading) 2026-10-16 — The final sentence claims that only the constant term survives in the limit, which is not justified. The solution does not actually use the Taylor expansions of cos x and sin x to compute the limit, so a student would learn an incorrect reasoning method.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The Maclaurin series for 1 - cos(x) is incorrectly stated as starting with x^2/2 (it should be x^2/2 - x^4/24 + ...), and the sign is flipped. Furthermore, the solution ignores the denominator x*sin(x) in the limit evaluation, claiming only the numerator's constant term survives, which is mathematically invalid for a 0/0 indeterminate form.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.