Answer
$\frac{1}{5}$
Work Step by Step
The original limit is of an indeterminate form $\frac{0}{0}$. Hence we use L'Hopital's:
$\displaystyle\lim_{x\to0}\frac{\sin(\pi x)}{\sin(5\pi x)}=\lim_{x\to0}\frac{\pi\cos(\pi x)}{5\pi\cos(5\pi x)}=\frac{\pi}{5\pi}=\fbox{$\frac{1}{5}$}$
