Question

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\frac{\mathrm{\pi }}{2}-x\right)\tan x$

Answer 1

$$\begin{gathered} \underset{x\to 1}{\lim }\frac{1-{\displaystyle \frac{1}{x}}}{\sin \mathrm{\pi }\left(x-1\right)} \\ =\underset{x\to 1}{\lim }\frac{x-1}{x\sin \mathrm{\pi }\left(x-1\right)} \\ \mathrm{Let}y=x-1 \\ x\to 1 \\ \therefore y\to 0 \\ =\underset{y\to 0}{\lim }\frac{y}{\left(y+1\right)\sin \mathrm{\pi }y} \\ =\underset{y\to 0}{\lim }\frac{1}{\mathrm{\pi }\left(y+1\right)\times {\displaystyle \frac{\sin \mathrm{\pi }y}{\mathrm{\pi }y}}} \\ =\frac{1}{\mathrm{\pi }\left(0+1\right)\times 1} \\ =\frac{1}{\mathrm{\pi }} \end{gathered}$$