Question

Evaluate the following limits in
$\underset{x\to \frac{\pi }{2}}{\text{lim}}\frac{\text{tan 2x}}{x-\frac{\pi }{2}}$

Answer 1

Here $\underset{x\to \frac{\pi }{2}}{\text{lim}}\frac{\text{tan 2x}}{x-\frac{\pi }{2}}\left[\frac{0}{0}\text{form}\right]$

Put x = $\frac{\pi }{2}$+ y as $x\to \frac{\pi }{2},y\to 0$

$\therefore \underset{y\to 0}{\text{lim}}\frac{\text{tan 2}(\frac{\pi }{2}+y)}{\frac{\pi }{2}+y-\frac{\pi }{2}}\underset{y\to 0}{\text{lim}}\frac{\text{tan}(\pi +2y)}{y}$

= $\underset{y\to 0}{\text{lim}}\frac{\text{tan}2y}{y}$

= $\underset{y\to 0}{\text{lim}}\frac{\text{tan}2y}{2y}\times 2$

= 1 $\times$ 2 = 2.