Limit w/Tangent: Solve & Discuss

[dobry_den]

Homework Statement

Solve the following limit:

\lim_{x \rightarrow 1}(1-x)\tan{\left(\frac{\pi x}{2}\right)}

The Attempt at a Solution

I solved it using L'Hospital rule, it's equal to 2/pi, but is there any other way how to solve it? thanks a lot!

The same question would apply to

\lim_{x \rightarrow \frac{\pi}{3}}\left(\frac{\sin{\left(x-\frac{\pi}{3}\right)}}{1-2\cos{x}}\right)

 

[dobry_den]

Well, I've finally found a different way how to solve it... I post it here since it can be useful to someone..

I substituted x with y = x-1, and then after some steps I've arrived at the following limit:

\lim_{y \rightarrow 0}\frac{\frac{\pi}{2}y}{\sin{\frac{\pi}{2}y}}\frac{2}{\pi}\cos{\frac{\pi}{2}y} = \frac{2}{\pi}