Question

Evaluate the following limits.
$\underset{x\to \frac{\pi }{2}}{lim}\frac{\cos x}{x-\left(\frac{\pi }{2}\right)}$.

Answer 1

$\begin{array}{c}{lim}_{x\to \frac{\pi }{2}}\frac{\cos x}{x-\frac{\pi }{2}}\\ Itis\frac{0}{0}form\\ UseLHospitalrule\\ \Rightarrow {lim}_{x\to \frac{\pi }{2}}\frac{-\sin x}{1-0}\\ \Rightarrow {lim}_{x\to \frac{\pi }{2}}\frac{-\sin \frac{\pi }{2}}{1}=-1Ans.\end{array}$