\[ \begin{split} \lim_{x \to 0}x\operatorname{ctg} 2x= \lim_{x \to 0}\frac{2x\cdot \cos 2x}{2\sin 2x}=\\[6pt] =\frac{1}{2}\lim_{n \to 0}\frac{2x}{\sin 2x}\cdot \cos 2x=\frac{1}{2} \end{split} \]