Question

$\underset{x\to 2}{lim}\frac{\sqrt{1-cos2(x-2)}}{x-2}$ is equal to

• A. Exist and equals $\sqrt{2}$
• B. Exists and is equal to $-\sqrt{2}$
• C. Does not exist
• D. Does not exist because L.H.L is not equal to R.H.L.

Answer 1

The correct option is C Does not exist

$\underset{x\to 2}{lim}\frac{\sqrt{1-cos2(x-2)}}{x-2}$

$1-cos2x=2si{n}^{2}x$

$\underset{x\to 2}{lim}\frac{\sqrt{2si{n}^2(x-2)}}{x-2}$

$\underset{x\to 2}{lim}\frac{\sqrt{2}|sin(x-2)|}{x-2}$ use

$\Rightarrow \underset{x\to 2^{+}}{lim}\frac{\sqrt{2}sin(x-2)}{x-2}=\sqrt{2}$

$\Rightarrow \underset{x\to 2^{-}}{lim}\frac{\sqrt{2}sin(x-2)}{x-2}=-\sqrt{2}$

so $LHL\ne RHL$