Question

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

• A. $\frac{\pi}{2}$
• B. $\pi$
• C. $\frac{2}{\pi}$
• D. 0

Answer 1

Answer: (C)

$\frac{2}{\pi}$

Answer 2

$\lim_{x \rightarrow 1}(1 - x)\tan\left( \frac{\pi x}{2} \right)$, Put $1 - x = y \Rightarrow$ as $x \rightarrow 1,y \rightarrow 0$

Thus $\lim_{y \rightarrow 0}y$ tan $\frac{\pi(1 - y)}{2} = \lim_{y \rightarrow 0}\frac{2}{\pi}.\frac{\left( \frac{\pi y}{2} \right)}{\tan\left( \frac{\pi y}{2} \right)}$ $= \frac{2}{\pi} \times 1 = \frac{2}{\pi}$.