# Homework Help: Limit with tangent

1. Jan 25, 2008

### dobry_den

1. The problem statement, all variables and given/known data

Solve the following limit:

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

3. 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

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

Last edited: Jan 25, 2008
2. Jan 27, 2008

### 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:

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