Question

The value of $\underset{x\to 1}{lim}\frac{\log x}{\sin \pi x}$ is

Answer 1

$l={lim}_{x\to 1}\frac{logx}{sin\left(\pi 21\right)}$ $(\frac{0}{0}$ form)

By L'Hospital rule

$l={lim}_{x\to 1}\frac{\frac{1}{x}}{\pi cos\left(\pi x\right)}$

$l={lim}_{x\to 1}\frac{1}{\pi xcos\left(\pi x\right)}$

$l=-\frac{1}{\pi }$