Question

lim x-0 x/tan(2x)

Answer 1

\[\begin{align*}\lim_{x\to0}\frac{x}{\tan2x}&=\lim_{x\to0}\frac{x\cos2x}{\sin2x}\\
&=\frac{1}{2}\left(\lim_{x\to0}\frac{2x}{\sin2x}\right)\left(\lim_{x\to0}{\cos2x}\right)\\
&=\frac{1}{2}\cdot1\cdot1\\
&=\frac{1}{2}
\end{align*}\]