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

$$\begin{gathered} \underset{\mathrm{x}\to 0}{\mathrm{Lt}}\frac{\sqrt{1-\cos 2\mathrm{x}}}{\sqrt{2}\mathrm{x}} \\ \underset{\mathrm{x}\to 0}{\mathrm{Lt}}\frac{\sqrt{2}\left|\sin \mathrm{x}\right|}{\sqrt{2}\mathrm{x}} \\ \mathrm{LHL}=\underset{\mathrm{x}\to 0}{\mathrm{Lt}}\frac{-\sin \mathrm{x}}{\mathrm{x}}=-1 \\ \mathrm{RHL}=\underset{\mathrm{x}\to 0}{\mathrm{Lt}}\frac{\sin \mathrm{x}}{\mathrm{x}}=1 \\ \mathrm{LHL}\ne \mathrm{RHL} \\ \mathrm{Limit} \mathrm{does} \mathrm{not} \mathrm{exist} \end{gathered}$$