Question

$\underset{h\to 0}{lim}\frac{\sin \sqrt{x+h}-\sin \sqrt{x}}{h}=$__________.

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

Answer 1

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

$L{t}_{x\to 0}\frac{\sin \sqrt{x+h}-\sin \sqrt{x}}{(x+h)-x}$

Let $f\left(x\right)=\sin \sqrt{x}$

$L{t}_{h\to 0}\frac{\sin x+h-\sin \sqrt{x}}{(x+h)-x}=L{t}_{x\to 0}f\frac{(x+h)-f\left(x\right)}{h}=f^1\left(x\right)\Rightarrow \cos \sqrt{x}\times \frac{1}{2\sqrt{x}}\Rightarrow \frac{\cos \sqrt{x}}{2\sqrt{x}}$