Question

$\underset{h\to 0}{\lim }\frac{\sqrt{x+h}-\sqrt{x}}{h},x\ne 0$

Answer 1

$\underset{h\to 0}{\lim }\left[\frac{\sqrt{x+h}-\sqrt{x}}{h}\right]$

It is of the form $\frac{0}{0}$.
Rationalising the numerator:
= $\underset{h\to 0}{\lim }\left[\frac{\left(\sqrt{x+h}-\sqrt{x}\right)}{h}\times \frac{\left(\sqrt{x+h}+\sqrt{x}\right)}{\left(\sqrt{x+h}+\sqrt{x}\right)}\right]$

= $\underset{h\to 0}{\lim }\left[\frac{\left(x+h\right)-x}{h\left(\sqrt{x+h}+\sqrt{x}\right)}\right]$

= $\frac{1}{\sqrt{x}+\sqrt{x}}$

= $\frac{1}{2\sqrt{x}}$