Question

Find the following limit:
$\underset{x\to 0}{lim}\frac{1-\cos x}{x^2}.$

Answer 1

$\underset{x\to 0}{lim}\frac{1-\cos x}{x^2}$

$=\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{x^2}$

$=2\underset{x\to 0}{lim}\frac{\sin \frac{x}{2}}{\frac{x}{2}}\cdot \frac{\sin \frac{x}{2}}{\frac{x}{2}}\times \frac{1}{4}\underset{x\to 0}{lim}\frac{\sin \frac{x}{2}}{\frac{x}{2}}=1$

$=2\times \frac{1}{4}$

$=\frac{1}{2}$