Answer
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = (2\sqrt{a})~f'(a)$
Work Step by Step
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = \lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} \cdot \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}}$
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = \lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a} \cdot (\sqrt{x}+\sqrt{a})$
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = \lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a} \cdot \lim\limits_{x \to a}(\sqrt{x}+\sqrt{a})$
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = f'(a) \cdot \lim\limits_{x \to a}(\sqrt{x}+\sqrt{a})$
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = f'(a) \cdot (\sqrt{a}+\sqrt{a})$
$\lim\limits_{x \to a} \frac{f(x)-f(a)}{\sqrt{x}-\sqrt{a}} = (2\sqrt{a})~f'(a)$
