Question

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

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\frac{\sin \sqrt{x}-\sin \sqrt{a}}{x-a} \\ =\underset{x\to a}{\lim }\frac{2\cos \left({\displaystyle \frac{\sqrt{x}+\sqrt{a}}{2}}\right)\sin \left({\displaystyle \frac{\sqrt{x}-\sqrt{a}}{2}}\right)}{\left(\sqrt{x}+\sqrt{a}\right)\left(\sqrt{x}-\sqrt{a}\right)} \\ =\underset{x\to a}{\lim }\frac{2\cos \left({\displaystyle \frac{\sqrt{x}+\sqrt{a}}{2}}\right)}{\sqrt{x}+\sqrt{a}}\times \frac{\sin \left({\displaystyle \frac{\sqrt{x}-\sqrt{a}}{2}}\right)}{2\left({\displaystyle \frac{\sqrt{x}-\sqrt{a}}{2}}\right)} \\ =\frac{1}{\sqrt{a}+\sqrt{a}}\cos \left(\frac{2\sqrt{a}}{2}\right) \\ \Rightarrow \frac{1}{2\sqrt{a}}\cos \sqrt{a} \end{gathered}$$