Question

The value of $\underset{x\to 1}{lim}{(2-x)}^{\tan \left(\frac{\pi x}{2}\right)}$ is

• A. $e^{-2/\pi }$
• B. $e^{1/\pi }$
• C. $e^{2/\pi }$
• D. $e^{-1/\pi }$

Answer 1

The correct option is C $e^{2/\pi }$

$\underset{x\to 1}{lim}(2-x{)}^{\tan \frac{\pi x}{2}}=\underset{x\to 1}{lim}{\{1+(1-x\left)\right\}}^{\tan \frac{\pi x}{2}}$

$=e^{\underset{x\to 1}{lim}(1-x)\tan \frac{\pi x}{2}}$

$=e^{\underset{x\to 1}{lim}(1-x)cot\left(\frac{\pi }{2}-\left(\frac{\pi x}{2}\right)\right)}$

$=e^{\underset{x\to 1}{lim}\frac{(1-x)}{\tan \left(\frac{\pi }{2}-\frac{\pi x}{2}\right)}}$

$=e^{\frac{2}{\pi }\underset{x\to 1}{lim}\frac{\frac{\pi }{2}(1-x)}{\tan \left(\frac{\pi }{2}(1-x)\right)}}$

$=e^{2/\pi }$