Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x$

 

Solution:

$=\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x$

$=-1 \times \lim _{y \rightarrow 0} y \tan \left(y+\frac{\pi}{2}\right)\left[x-\frac{\pi}{2}=y\right]$

$=-1 \times \lim _{y \rightarrow 0} y \cot y \times-1$

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

$=1$

$\therefore \lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x=1$

 

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