$\underset{x\to 3}{\lim }\frac{\sqrt{1-\cos 2(x-3)}}{x-3}$

$1-\cos 2(x-3)=2{\sin }^2(x-3)$

$\varphi t\sqrt{1-\cos 2(x-3)}=\sqrt{2}\left|\sin \right(x-3\left)\right|$

$\text{ So, }\underset{x\to 3+}{\lim }\frac{\sqrt{1-\cos 2(x-3)}}{x-3}=\sqrt{2}\underset{x\to 3+}{\lim }\frac{\sin (x-3)}{x-3}=\sqrt{2}$

whereas

$\underset{x\to 3-}{\lim }\frac{\sqrt{1-\cos 2(x-3)}}{x-3}=\sqrt{2}\underset{x\to 3-}{\lim }\frac{-\sin (x-3)}{x-3}=-\sqrt{2}$