Answer
$\lim\limits_{x \to 1}(2-x)^{tan(\pi~x/2)} = e^{(2/\pi)}$
Work Step by Step
Let $~~y = \lim\limits_{x \to 1}(2-x)^{tan(\pi~x/2)}$
$ln~y = \lim\limits_{x \to 1}ln~(2-x)^{tan(\pi~x/2)}$
$ln~y = \lim\limits_{x \to 1}tan(\frac{\pi~x}{2})~ln~(2-x)$
$ln~y = \lim\limits_{x \to 1}~\frac{ln~(2-x)}{cot(\frac{\pi~x}{2})}$
$ln~y = \lim\limits_{x \to 1}~\frac{\frac{-1}{2-x}}{\frac{-\pi}{2sin^2(\frac{\pi~x}{2})}}$
$ln~y = \lim\limits_{x \to 1}~\frac{2sin^2(\frac{\pi~x}{2})}{(\pi)(2-x)}$
$ln~y = \frac{2}{\pi}$
$y = e^{(2/\pi)}$
