Question

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

Answer 1

$\underset{x\to a}{lim}\frac{\sin \sqrt{x}-\sin \sqrt{a}}{(x-a)}\Rightarrow \%$ we use $L^{\prime }$ hosptals rule.

$\underset{x\to a}{lim}\left[\frac{\cos \sqrt{x}\times \frac{1}{2\sqrt{x}}}{1}\right]\Rightarrow \cos \sqrt{a}\times \frac{1}{2\sqrt{a}}$

$\therefore \underset{x\to a}{lim}\frac{[\sin \sqrt{x}-\sin \sqrt{a}]}{(x-a)}=\frac{1}{2\sqrt{a}}\cos \sqrt{a}$