Question

$\underset{x\to 0}{Lt}\frac{\sqrt{\frac{1}{2}(1-\cos x)}}{|x|}=$

• A. $1/2$
• B. $-1/2$
• C. $0$
• D. Does not exist

Answer 1

The correct option is D Does not exist

$\underset{x\to 0}{lt}\frac{\sqrt{\frac{1}{2}(1-\cos x)}}{|x|}=\frac{\sqrt{\frac{1}{2}(1-(1-2{\sin }^2\frac{x}{2}))}}{|x|}$

$=\frac{\sqrt{\frac{1}{2}(\text{/}1-\text{/}1+2{\sin }^2\frac{x}{2})}}{|x|}$

$=\frac{\sin \frac{x}{2}}{|x|}$

$\underset{x\to 0^{+}}{lt}=\frac{\sin \frac{x}{2}}{x}\Rightarrow \frac{\frac{x}{2}}{x}=\frac{1}{2}$

$\underset{x\to 0^{-}}{lt}=\frac{\sin \frac{x}{2}}{-x}=\frac{+\frac{x}{2}}{-x}=-\frac{1}{2}$

$\therefore$ Limit doesn't exists.