The value of $\underset{x\to b}{lim} \frac{\sqrt{x-a}-\sqrt{b-a}}{x^2-b^2},$for $b>a$

Solution:

 We have, 

$\begin{array}{l}\underset{x\to b}{lim} \frac{\sqrt{x-a}-\sqrt{b-a}}{x^2-b^2}\\ =\underset{x\to b}{lim} \frac{x-b}{(x-b)(x+b)\{\sqrt{x-a}+\sqrt{b-a}\}}\\ =\underset{x\to b}{lim} \frac{1}{(x+b)\{\sqrt{x-a}+\sqrt{b-a}\}}=\frac{1}{2b\left\{2\sqrt{b-a}\right\}}=\frac{1}{4b\sqrt{b-a}}\end{array}$