Question

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

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

Answer 1

The correct option is A $\frac{\cos \sqrt{x}}{2\sqrt{x}}$

Let, $L=\underset{h\to 0}{lim}\frac{sin\sqrt{x+h}-sin\sqrt{x}}{h}$
Clearly form of the limit is $\frac{0}{0}$

Thus using L Hospital's rule (differentiating numerator & denominator w.r.t h), we get

$L=\underset{h\to 0}{lim}\frac{\cos \sqrt{x+h}.\frac{1}{2\sqrt{x+h}}}{1}$

$L=\frac{\cos \sqrt{x}}{2\sqrt{x}}$