Exercise · ZW-LHOPITALS-RULE-013 · ★☆☆ Basic
Compute \(\displaystyle\lim_{x\to0}\frac{1-\cos x}{x^2}\).
Choose one option.
Show answer
- B. \(\tfrac12\)
Explanation
It is \(0/0\). Once: \(\frac{\sin x}{2x}\), again \(0/0\). Once more: \(\frac{\cos x}{2}\to\frac12\).
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-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).