Multiple applications of the rule

In the following example, we have to use l'H^opital's rule twice before we get an answer.

Example

Evaluate

$\lim_{x\rightarrow \pi}\frac{1+cos x}{(x=\pi)^2}$ Applying l’Hopital’s rule gives

$\frac{-sin x}{2(x-\pi)}$

which still produces 0/0 when we plug in $x=\pi$ . Going again, we get $\frac{-cos x}{2}=\frac{1}{2}$