Question

$\underset{h\to 0}{lim}\frac{\sqrt{x+h}-\sqrt{x}}{h}$ is equal to

• A. $\sqrt{x}$
• B. $\frac{1}{2\sqrt{x}}$
• C. $2\sqrt{x}$
• D. $\frac{1}{\sqrt{x}}$

Answer 1

The correct option is B $\frac{1}{2\sqrt{x}}$

The given limit can be written as,

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

We know that,

$\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

So, the given limit is,

$=\frac{1}{2}{x}^{(\frac{1}{2}-1)}=\frac{1}{2\sqrt{x}}$