Limits using Taylor series
Problem 7.677 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 \cos{\left(x \right)} - 2}{x \sin{\left(x \right)}} \).
- 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
- \[ \lim_{x \to 0}\left(\frac{2 \cos{\left(x \right)} - 2}{x \sin{\left(x \right)}}\right) = -1 \]Let x → 0: only the constant term survives.✓ Proved
Answer \( -1 \)
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 explanation in sentence 3 incorrectly states that only the constant term survives; the limit requires using the leading non‑zero terms of the Taylor expansions of numerator and denominator.qwen3.6:27b-mlx: fail (misleading) — The solution claims the limit is -1, but the correct value is -1/2. The reasoning in step 3 is flawed because it ignores the expansion of the denominator x*sin(x) ~ x^2, leading to a ratio of -2/2 = -1/2, not -1.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution claims the limit is -1, but the correct value is -1/2. The reasoning in step 3 is flawed because it ignores the expansion of the denominator x*sin(x) ~ x^2, leading to a ratio of -2/2 = -1/2, not -1.gpt-oss:20b: fail (misleading) 2026-10-18 — The explanation in sentence 3 incorrectly states that only the constant term survives; the limit requires using the leading non‑zero terms of the Taylor expansions of numerator and denominator.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 + ...), and the limit evaluation in step 3 ignores the denominator's expansion, leading to an incorrect result of -1 instead of -1/3.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The Maclaurin series for 1 - cos(x) is stated incorrectly as -x**4/24 + x**2/2; the correct expansion is x**2/2 - x**4/24 + ... (the signs are swapped for the x^4 term, though the x^2 term is correct). Furthermore, the solution claims the limit is -1, but using the correct expansion (2*(x^2/2)/x^2 = 1), the limit is actually 1. The stated answer and the derivation are both wrong.gpt-oss:20b: fail (misleading) 2026-10-15 — The final sentence incorrectly claims that only the constant term survives; in fact the leading non‑zero terms of the numerator and denominator determine the limit, yielding –1. This misleads a student about how the limit is obtained.
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.