Question

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

Answer 1

Using L' Hospital to $\underset{x\to 0}{lim}\frac{1-\sqrt{\cos x}}{x^2}$

Differentiating numerator & denominator $=\underset{x\to 0}{lim}\frac{\frac{-1(-\sin x)}{2\sqrt{cosx}}}{2x}$

$\Rightarrow \frac{\frac{\sin x}{2\sqrt{\cos x}}}{2x}$

$\Rightarrow \underset{x\to 0}{lim}\frac{\sin x}{2x\times 2\sqrt{\cos x}}$

$1$ as $\underset{x\to 0}{lim}$

$\Rightarrow \frac{1}{4\sqrt{\cos 0}}$

$=\frac{1}{4}$.