Question

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

Answer 1

${lim}_{x\to \frac{\pi }{2}}\frac{\cos x}{x-\left(\frac{\pi }{2}\right)}$ (we have a limit of $\frac{0}{0}$ form).

$\therefore$ Employing the L'Hospital Rule

${lim}_{x\to \frac{\pi }{2}}\frac{\cos x}{x-\frac{\pi }{2}}={lim}_{x\to \frac{\pi }{2}}\frac{-\sin x}{1}=-\sin \frac{\pi }{2}=-1$