Question

Find the value of $\underset{x\to \frac{\pi }{2}}{lim}\frac{\tan 2x}{x-\frac{\pi }{2}}$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\tan 2x}{x-\frac{\pi }{2}}$
Put $x-\frac{\pi }{2}=y$ so that $x\to \frac{\pi }{2},y\to 0$
$\therefore \underset{x\to \frac{\pi }{2}}{lim}\frac{\tan 2x}{x-\frac{\pi }{2}}=\underset{y\to 0}{lim}\frac{\tan 2(y+\frac{\pi }{2})}{y}$
$=\underset{y\to 0}{lim}\frac{\tan (\pi +2y)}{y}$
$=\underset{y\to 0}{lim}\frac{\tan 2y}{y}$
$=\underset{y\to 0}{lim}\frac{\tan 2y}{2y}\cdot 2=\left(1\right)\cdot 2=2$