Exercise · ZW-LHOPITALS-RULE-003 · ★★★ Advanced
Compute \(\lim_{x\to 0}\,\frac{x - \sin {x}}{x^{3}}\).
Choose one option.
Show answer
- A. \(\frac{1}{6}\)
Explanation
Substituting gives the indeterminate form 0/0, so L'Hôpital's rule applies by differentiating numerator and denominator: \(\frac{1 - \cos {x}}{3 x^{2}}\) → \(\frac{\sin {x}}{6 x}\) → \(\frac{\cos {x}}{6}\). The resulting limit is \(\frac{1}{6}\).
Review the concept: L'Hôpital's rule
Rule that computes limits of quotients in indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) by differentiating numerator and denominator separately.
\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]
More practice problems: l'hôpital's rule
- ZW-LHOPITALS-RULE-001 Compute lim _x→ 0 (- x + e^x - 1)/(x^2).
- ZW-LHOPITALS-RULE-002 Compute lim _x→ ∞ (ln (x ))/(x).
- ZW-LHOPITALS-RULE-004 Compute lim _x→ 0 (e^2 x - 1)/(3 x).
- ZW-LHOPITALS-RULE-005 Compute lim _x→ ∞ x^2 e^- x.
- ZW-LHOPITALS-RULE-006 Compute lim _x→ 1 (ln (x ))/(x - 1).
- ZW-LHOPITALS-RULE-007 Compute lim _x→ 0 (sin (3 x ))/(x).
- ZW-LHOPITALS-RULE-008 Compute lim _x→ 2 (x^2 - 4)/(x - 2).
- ZW-LHOPITALS-RULE-009 Compute lim _x→ 0 (1 - cos x)/(x^2).