Question

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

Answer 1

${lim}_{x\to a}\frac{\cos \sqrt{x}-\cos \sqrt{a}}{x-a}=\frac{0}{0}$ form

Using L-Hospital's rule

${lim}_{x\to a}\frac{(-\sin \sqrt{x})\times \frac{1}{2\sqrt{x}}}{1-0}-0$

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