differentiate f(x) using L'hopital and other problem 
*

*Evaluate: $\ \ \ \ \lim_{x\to1}(2-x)^{\tan(\frac{\pi}{2}x)}$

*Show that  the  inequality holds: $\ \ \ x^\alpha\leq \alpha x + (1-\alpha)\ \ \ (x\geq0, \,0<\alpha <1)$
Please help me with these. 
Either a hint or a full proof will do.
Thanks.

For the first problem: another useful method is to log the function to get $\frac{\log(2-x)}{\frac{1}{\tan(\frac{\pi x}{2})}}$, now you can apply L'Hospital's rule, take the limit and exponentiate it. I keep getting $e^{\frac{\pi}{2}}$, but you need to check the algebra more thoroughly
