Question

. $\underset{x\to 2}{lim}\left(\frac{\sqrt{1-\cos \left\{2\right(x-2\left)\right\}}}{x-2}\right)$

• A. does not exist
• B. equals $\sqrt{2}$
• C. equals $-\sqrt{2}$
• D. equals $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A)

does not exist

Let $x-2=t$
$\underset{t\to 0}{lim}\frac{\sqrt{1-\cos 2t}}{t}$
$=\underset{t\to 0}{lim}\sqrt{2}\frac{|\sin t|}{t}$
Clearly R.H.L. $=\sqrt{2}$

L.H.L. $=-\sqrt{2}$
Since R.H.L. $\ne$ L.H.L. So, limit does not exist.
Hence, option 'A' is correct.