\begin{equation*}
\begin{split}
& \lim_{x\rightarrow 0} \frac{2x \:\tan4x}{\cos2x-1} \\\\
& \lim_{x\rightarrow 0} \frac{2x \:\tan4x}{1 - 2 \sin^2 x -1} \\\\
& \lim_{x\rightarrow 0} \frac{2x \:\tan4x}{- 2 \sin^2 x} \\\\
& \lim_{x\rightarrow 0} \frac{2x}{-2 \sin x} \:.\: \frac {\tan4x}{\sin x} \\\\
& -1 \:.\: 4 \\\\
& -4
\end{split}
\end{equation*}
