Question

lim x-->pi (1-sen(x/2))/(pi-x)

Answer 1

$\lim _{x\to \:\pi }\left(\frac{1-\sin \left(\frac{x}{2}\right)}{\pi -x}\right)$
Se aplica la regla de L´Hopital:
$=\lim _{x\to \:\pi }\left(\frac{-\frac{1}{2}\cos \left(\frac{x}{2}\right)}{-1}\right)$
Se simplifica:
$=\lim _{x\to \:\pi }\left(\frac{\cos \left(\frac{x}{2}\right)}{2}\right)$
Se sutituye la variable:
$=\frac{\cos \left(\frac{\pi }{2}\right)}{2}=0$