Lim $x\to  0$ of $\frac{\sin(\pi x)}{\tan(\sqrt{3}x)}$ $$
   \lim_{x \to 0} \frac{\sin(\pi x)}{\tan(\sqrt{3} x)}
$$
I need a step by step explanation. Thank you.

Results going to be used:


*

*$\displaystyle \lim_{x \to 0} \frac{\sin{x}}{x} =1 = \lim_{x \to 0} \frac{\tan{x}}{x}$


What you have is 
\begin{align*}
\lim_{x \to 0}\: \frac{\sin(\pi{x})}{\tan\sqrt{3}{x}} &= \lim_{x \to 0} \: \frac{\sin\pi{x}}{\pi{x}} \times \frac{\sqrt{3}x}{\tan{\sqrt{3}x}} \times \frac{\pi{x}}{\sqrt{3}x} \\ &= \lim_{x \to 0} \: \frac{\pi{x}}{\sqrt{3}x} = \frac{\pi}{\sqrt{3}}
\end{align*}
