Question

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

• A. $1$
• B. $-1$
• C. $0$
• D. does not exist

Answer 1

The correct option is D does not exist

$\underset{x\to 0}{lim}\frac{\sqrt{\frac{1}{2}(1-\cos x)}}{x}=\underset{x\to 0}{lim}\frac{\sqrt{\frac{1}{2}\times 2{\sin }^2\left(\frac{x}{2}\right)}}{x}$

$=\underset{x\to 0}{lim}\pm \frac{\sin \left(x/2\right)}{2\times \frac{x}{2}}$

RHL

$=\frac{1}{2}\underset{x\to 0^{+}}{lim}\frac{\sin \left(x/2\right)}{\frac{x}{2}}$

as $x\to 0^{+}$

$\sin x>0$

$x/2\to 0$

$=\frac{1}{2}\times 1=\frac{1}{2}$

LHL

$=\frac{1}{2}\underset{x\to 0^{-}}{lim}\frac{-\sin \left(x/2\right)}{\frac{x}{2}}$ as $x\to 0^{-};\sin x<0$

$x/2\to 0^{-}$

$=-\frac{1}{2}$

LHL$\ne$RHL $\Rightarrow$Limit does not exist