Question

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

Answer 1

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

put $x=a$ we get an undetermined form of $\frac{0}{0}$

so apply L'hospital rule

we get $\underset{x\to a}{lim}\frac{\frac{-sin\sqrt{x}}{2\sqrt{x}}}{1}$

now put $x=a$

$=\frac{-sin\sqrt{a}}{2\sqrt{a}}$