Limits using Taylor series
Problem 7.706 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 - 3 \cos{\left(x \right)}}{x \sin{\left(x \right)}} \).
- 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
- \[ \lim_{x \to 0}\left(\frac{3 - 3 \cos{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \frac{3}{2} \]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: passqwen3.6:27b-mlx: fail (error) — The solution claims the limit is 3/2, but the correct limit is 1. The expansion of 1 - cos(x) starts with x^2/2, and x*sin(x) starts with x^2, so the ratio approaches 1. The provided solution incorrectly states the final limit is 3/2.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution claims the limit is 3/2, but the correct limit is 1. The expansion of 1 - cos(x) starts with x^2/2, and x*sin(x) starts with x^2, so the ratio approaches 1. The provided solution incorrectly states the final limit is 3/2.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims that 'only the constant term survives' when evaluating the limit, which is mathematically incorrect reasoning for a 0/0 indeterminate form. The limit is determined by the ratio of the leading non-zero terms (x^2/x^2), not just a constant term. Additionally, the expansion in step 1 is listed in descending order of powers, which is non-standard and confusing.gpt-oss:20b: fail (misleading) 2026-10-16 — The final sentence claims the limit is 3/2 by saying “only the constant term survives,” but it does not actually apply the Taylor expansion or show how the terms cancel. A student could be misled into thinking the limit follows directly from a constant term, which is not justified here.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly states the Maclaurin series for 1 - cos(x) as starting with x^2/2, whereas the correct expansion is x^2/2 - x^4/24 + ... (the signs are reversed). Furthermore, the limit evaluation in step 3 is asserted without showing the necessary division by the series expansion of x*sin(x) ~ x^2, making the logic incomplete and the intermediate algebra wrong.gpt-oss:20b: pass 2026-10-16 — The solution’s setup and algebra are correct; the final sentence could be clearer by explicitly showing the series substitution rather than the vague “only the constant term survives.”
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.