\begin{align*}

\lim\limits_{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right) & =\frac{\sin\left(2\cdot \frac{\pi }{3}\right)}{\sin\:\left(\frac{\pi }{3}\right)} \\

& =\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}\\

& =1 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\

\end{align*}