Question

Evaluate the limit
$\underset{x\to 1}{lim}\frac{1-\frac{1}{x}}{\sin \pi \left(x-1\right)}$

Answer 1

${lim}_{x\to 1}\frac{1-\frac{1}{x}}{\sin \pi (x-1)}$

it is of the form $\frac{0}{0}$

So, we will apply L Hospital rule until we will get determinate form

Differentiating numerator and denominator we will get,

${lim}_{x\to 1}\frac{-(-\frac{1}{x^2})}{\cos \pi (x-1)}=\frac{1}{1}=1$