Question

Solve:
$\underset{x\to \frac{\pi }{2}}{lim}\frac{\cot x}{x-\pi /2}$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\cot x}{x-\frac{\pi }{2}}$

$\underset{h\to 0}{lim}\frac{\cot \frac{\pi }{2}-h}{-h}$

$-\underset{h\to 0}{lim}\frac{\tan h}{h}=1$

Put $h=\frac{\pi }{2}-x$

i.e., $x=\frac{\pi }{2}-h$

as $x\to \frac{\pi }{2}h\to 0$

$(\therefore \underset{x\to 0}{lim}\frac{tanx}{x}=1)$.