The value of ${\lim }_{x\to 2}\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}$:

Let $t=x-2$. As $x\to 2,t\to 0$.The limit is ${\lim }_{t\to 0}\frac{\sqrt{1-\cos 2t}}{t}$.Since $1-\cos 2t=2{\sin }^{2}t$, the numerator is $\sqrt{2{\sin }^{2}t}=\sqrt{2}|\sin t|$.Limit $={\lim }_{t\to 0}\frac{\sqrt{2}|\sin t|}{t}$.Right hand limit ($t\to 0^{+}$): $\frac{\sqrt{2}\sin t}{t}\to \sqrt{2}$.Left hand limit ($t\to 0^{-}$): $\frac{\sqrt{2}(-\sin t)}{t}\to -\sqrt{2}$.Since RHL $\ne$ LHL, the limit does not exist.