Question

Evaluate $\underset{x\to a}{lim}\frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$

Answer 1

We have,

$\underset{x\to a}{lim}\frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$

This is $\frac{0}{0}$ form.

So, apply L-hospital rule,

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

$\underset{x\to a}{lim}2\sqrt{x}\cos x$

$=2\sqrt{a}\cos a$

Hence, this is the answer.