lim x approaches pi of (x-pi) / sinx

Question

the answer is supposedly 1?

But using l'hopitlals rule i got -1..

help please!

Answer 1

Let x=(pi)+h, where h is small so that the expression is h/(sin((pi)+h)=h/(sin(pi)cos(h)+hcos(pi)), because sinh=h when h is small. This becomes h/(-h)=-1 in the limit as h approaches zero and x approaches (pi). So L'Hôpital was right!