Question

If $f\left(x\right)=\frac{1-\cos (x-\pi )}{(\pi -x{)}^2},x\ne \pi$ is continuous at $x=0$, find $f\left(\pi \right)$.

Answer 1

$f\left(\pi \right)=\underset{x\to \pi }{lim}f\left(x\right)=\underset{x\to \pi }{lim}\frac{1-\cos (x-\pi )}{(\pi -x{)}^2}\left(\frac{0}{0}\right)$ form

Apply L'H Rule

$\underset{x\to \pi }{lim}\frac{(-1)\sin (x-\pi )}{2(\pi -x)}\left(\frac{0}{0}\right)$ again

$f\left(\pi \right)=\underset{x\to \pi }{lim}\frac{\cos (x-\pi )}{2}$

$=\frac{1}{2}$