Exercise · ZW-LHOPITALS-RULE-001 · ★★☆ Intermediate
Compute \(\lim_{x\to 0}\,\frac{- x + e^{x} - 1}{x^{2}}\).
Choose one option.
Show answer
- C. \(\frac{1}{2}\)
Explanation
Substituting gives the indeterminate form 0/0, so L'Hôpital's rule applies by differentiating numerator and denominator: \(\frac{e^{x} - 1}{2 x}\) → \(\frac{e^{x}}{2}\). The resulting limit is \(\frac{1}{2}\).
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-002 Compute lim _x→ ∞ (ln (x ))/(x).
- ZW-LHOPITALS-RULE-003 Compute lim _x→ 0 (x - sin x)/(x^3).
- 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).