Limits using Taylor series
Problem 7.601 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{\cos{\left(x \right)} - 1}{x \sin{\left(x \right)}} \).
- 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
- \[ \lim_{x \to 0}\left(\frac{\cos{\left(x \right)} - 1}{x \sin{\left(x \right)}}\right) = - \frac{1}{2} \]Let x → 0: only the constant term survives.✓ Proved
Answer \( - \frac{1}{2} \)
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 -1/2, but the correct value is -1/2. Wait, let me re-evaluate. cos(x) ~ 1 - x^2/2. sin(x) ~ x. Numerator ~ -x^2/2. Denominator ~ x^2. Ratio ~ -1/2. The answer is correct. However, the reasoning in line 3 is flawed. It says 'only the constant term survives' which is vague and potentially misleading if not explained properly (dividing series). But more importantly, line 1 states '1 - cos(x) = -x**4/24 + x**2/2 + ...'. This is correct. Line 2 checks the error term. Line 3 jumps to the answer. The logic is: (cos(x)-1)/(x sin(x)) = -(1-cos(x))/(x sin(x)). 1-cos(x) = x^2/2 - x^4/24 + ... x sin(x) = x(x - x^3/6 + ...) = x^2 - x^4/6 + ... Ratio = -(x^2/2 - ...)/(x^2 - ...) = -1/2. The solution skips the division step and just asserts the result. Is it an error? The sentence 'Let x -> 0: only the constant term survives' is a poor explanation of dividing power series. It implies substituting x=0 into the series terms directly, which would give 0/0. It fails to explain that one must divide the leading terms. This is misleading.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims the limit is -1/2, but the correct value is -1/2. Wait, let me re-evaluate. cos(x) ~ 1 - x^2/2. sin(x) ~ x. Numerator ~ -x^2/2. Denominator ~ x^2. Ratio ~ -1/2. The answer is correct. However, the reasoning in line 3 is flawed. It says 'only the constant term survives' which is vague and potentially misleading if not explained properly (dividing series). But more importantly, line 1 states '1 - cos(x) = -x**4/24 + x**2/2 + ...'. This is correct. Line 2 checks the error term. Line 3 jumps to the answer. The logic is: (cos(x)-1)/(x sin(x)) = -(1-cos(x))/(x sin(x)). 1-cos(x) = x^2/2 - x^4/24 + ... x sin(x) = x(x - x^3/6 + ...) = x^2 - x^4/6 + ... Ratio = -(x^2/2 - ...)/(x^2 - ...) = -1/2. The solution skips the division step and just asserts the result. Is it an error? The sentence 'Let x -> 0: only the constant term survives' is a poor explanation of dividing power series. It implies substituting x=0 into the series terms directly, which would give 0/0. It fails to explain that one must divide the leading terms. This is misleading.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly states the Maclaurin series for 1 - cos(x) as starting with x^2/2, but then uses a limit involving division by x^4 to justify the approximation, which is inconsistent and logically flawed for evaluating the given limit. The limit in line 2 is irrelevant to the problem statement and the conclusion in line 3 does not follow from the provided premises.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly expands the numerator as 1 - cos(x) instead of the required cos(x) - 1, introducing a sign error. Additionally, the intermediate limit in step 2 is irrelevant to the problem statement and does not logically lead to the final result.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.