Limit of \( \displaystyle \frac{x - \sin{\left(x \right)}}{x^{3}} \) as \( x \to 0 \)
Problem 1.34 · easy Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{x - \sin{\left(x \right)}}{x^{3}} \).
- \[ \lim_{x \to 0^+}\left(\frac{x - \sin{\left(x \right)}}{x^{3}}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x - \sin{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right) \]lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{3 x^{2}}\right) \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}{\frac{d}{d x} 3 x^{2}}\right) \]lhopitalThe limit is still an indeterminate form 0/0, so apply L'Hopital's rule again.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{6 x}\right) \]simplify algebraCompute the derivatives of the new numerator and denominator. Factor out the constant 1/6.✓ Proved
- \[ = \frac{\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)}{6} \]limit-lawApply the limit law for the product of two functions.✓ Proved
- \[ = \frac{1}{6} \]trig-limit simplifyUse the fundamental trigonometric limit sin(x)/x -> 1 as x approaches 0. Final evaluation of the limit.✓ Proved
Answer \( \frac{1}{6} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where Derivative(x**3, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(x**3, x) = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where Derivative(3*x**2, x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(3*x**2, x) = 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy took the limit from both sides and got the stated value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies L'Hôpital's rule twice and uses standard limit laws and trigonometric limits. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies L'Hopital's rule twice, handles the algebraic simplification and limit laws properly, and uses the standard trigonometric limit. Each step changes only one aspect of the expression and is labeled correctly.gpt-oss:20b: pass 2026-09-29
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
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.