Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$

 

Solution:

$=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$

As $x$ tends to $\pi / 2, x-\pi / 2$ tends to zero.

Let,

$\mathrm{y}=\mathrm{x}-\frac{\pi}{2}$

$=\lim _{y \rightarrow 0} \frac{\tan \left(2 y+\frac{\pi}{2} \times 2\right)}{y}$

$=\lim _{y \rightarrow 0} \frac{\tan (\pi+2 y)}{y}$

$=\lim _{y \rightarrow 0} \frac{\tan 2 y}{2 y} \times 2$

$=2$

$\therefore \lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}=2$

 

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