Question

Evaluate $\underset{h\to 0}{lim}\frac{\sqrt{(x+h)}-\sqrt{x}}{h}$ which is $=\frac{1}{a\sqrt{\left(x\right)}}$ Find $a$

Answer 1

Required limit is, $=\underset{h\to 0}{lim}\frac{\sqrt{(x+h)}-\sqrt{x}}{h}$
$=\underset{h\to 0}{lim}\frac{\sqrt{(x+h)}-\sqrt{x}}{h}.\frac{\sqrt{(x+h)}+\sqrt{x}}{\sqrt{(x+h)}+\sqrt{x}}$
$=\underset{h\to 0}{lim}\frac{(x+h)-x}{h\sqrt{(x+h)}+\sqrt{x}}$
$=\underset{h\to 0}{lim}\frac{1}{\sqrt{(x+h)}+\sqrt{x}}=\frac{1}{2\sqrt{x}}$.