\documentclass{article} \begin{document} \noindent Applying l'Hopital's rule, one has \[ \lim_{x\to0}\frac{\ln\sin\pi x}{\ln\sin x} = \lim_{x\to0}\frac{\pi\frac{\cos\pi x}{\sin\pi x}}{\frac{\cos x}{\sin x}} = \lim_{x\to0}\frac{\pi\tan x}{\tan\pi x} = \lim_{x\to0}\frac{\pi/\cos^2 x}{\pi/\cos^2 \pi x} = \lim_{x\to0}\frac{\cos^2\pi x}{\cos^2 x} = 1 \] \end{document}