\begin{equation*}
\begin{split}
& \lim_{x \rightarrow 0} \frac{\sin4x}{\sqrt{x+2}-\sqrt{2}} \quad {\color {blue} \times \frac {\sqrt{x+2} + \sqrt{2}}{\sqrt{x+2} + \sqrt{2}}} \\\\
& \lim_{x \rightarrow 0} \frac {\sin 4x \:.\: (\sqrt{x+2} + \sqrt{2})}{x + 2 - 2} \\\\
& \lim_{x \rightarrow 0} \frac{\sin 4x \:.\: (\sqrt{x+2} + \sqrt{2})}{x} \\\\
& \lim_{x \rightarrow 0} 4 \:.\: (\sqrt{x+2} + \sqrt{2}) \\\\
& 4 \:.\: (\sqrt{0+2} + \sqrt{2}) \\\\
& 4 \:.\: 2 \sqrt {2} \\\\
& 8 \sqrt{2}
\end{split}
\end{equation*}
