$\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$

solution:

Given $\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$

Now we have to rationalize the denominator by multiplying the dividing by its rationalizing factor then we get

$\Rightarrow$$\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}=\lim _{x \rightarrow a}\left[\frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}} \times \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}}\right]$

On simplifying and splitting the denominator we get

$\Rightarrow$$\lim _{x \rightarrow a}\left[\frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}} \times \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}}\right]=\lim _{x \rightarrow a} \frac{\sin x-\sin a}{x-a} \lim _{x \rightarrow a}(\sqrt{x}+\sqrt{a})$

Now as $\sin x-\sin a=2 \cos \left(\frac{x+a}{2}\right) \sin \left(\frac{x-a}{2}\right)$

Substituting this in above equation we get

$\Rightarrow$$\lim _{x \rightarrow a} \frac{\sin x-\sin a}{x-a} \lim _{x \rightarrow a}(\sqrt{x}+\sqrt{a}) \lim _{x \rightarrow a} \frac{2 \cos \left(\frac{x+a}{2}\right) \sin \left(\frac{x-a}{2}\right)}{x-a} \lim _{x \rightarrow a}(\sqrt{x}+\sqrt{a})$

$\Rightarrow$$\lim _{x \rightarrow a} \frac{2 \cos \left(\frac{x+a}{2}\right) \sin \left(\frac{x-a}{2}\right)}{x-a} \lim _{x \rightarrow a}(\sqrt{x}+\sqrt{a})=2 \sqrt{a} \lim _{x \rightarrow a} \frac{\sin \left(\frac{x-a}{2}\right)}{\frac{x-a}{2}} \lim _{x \rightarrow a} \cos \left(\frac{x+a}{2}\right)$

Now as $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

Applying the limits in above equation we get

$\Rightarrow$$2 \sqrt{a} \lim _{x \rightarrow a} \frac{\sin \left(\frac{x-a}{2}\right)}{\frac{x-a}{2}} \lim _{x \rightarrow a} \cos \left(\frac{x+a}{2}\right)=2 \sqrt{a} \cdot 1 \cdot \cos a$

$\Rightarrow$$\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}=2 \sqrt{a} \cos a$