Home›Calculus 2›Limits using Taylor series›Problem 7.679

Limits using Taylor series

Problem 7.679 · 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).
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{5}}{120} + \frac{x^{3}}{6} - x + \sin{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{x^{5}}{40} + \frac{x^{3}}{2}}{x^{3}} = \frac{1}{2} - \frac{x^{2}}{40} \]
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 \( \frac{1}{2} \)

Lines: 3 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
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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly states the Maclaurin series for -x + sin(x) as starting with x^5/120, omitting the crucial -x^3/6 term. This error is masked by the fact that the final answer is correct, but the reasoning in step 1 is mathematically false.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly states the Maclaurin series for -x + sin(x) as starting with x^5/120, omitting the crucial -x^3/6 term. This error is masked by the fact that the final answer is correct, but the reasoning in step 1 is mathematically false.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series for sin(x) and applies it to evaluate the limit. The intermediate steps are algebraically sound and the final result is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly states the Maclaurin series for -x + sin(x) as starting with x^5/120, omitting the leading -x^3/6 term. This algebraic error in the series expansion leads to an incorrect intermediate expression, although the final limit value happens to be correct due to a compensating error in the division step.
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 2026-10-18 with SymPy 1.14.0.