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

$$\begin{gathered} =\underset{x\to 1}{lim} (2-x{)}^{\tan \frac{\mathrm{\pi }x}{2}} \\ =\underset{x\to 1}{lim} \{1+(1-x){\}}^{\tan \frac{2x}{2}} \end{gathered}$$

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

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