Finding the Limit of a Complex Expression

[daniel_i_l]

Homework Statement

Find the limit:
$$lim_{x \rightarrow 1} (2-x)^{tan \frac{\pi x}{2}}$$

Homework Equations

The Attempt at a Solution

I first tried to find the limit of ln of the function inorder to turn the power into a multiplication and got:
$$lim_{x \rightarrow 1} \frac{ ln(2-x) sin \frac{\pi x}{2}}{cos \frac{\pi x}{2}}$$
Then I used l'hospital's rule and got:
$$lim_{x \rightarrow 1} ln( (2-x)^{tan \frac{\pi x}{2}} ) = \pi / 2$$
That means that $$lim_{x \rightarrow 1} (2-x)^{tan \frac{\pi x}{2}} = e^{\pi / 2}$$
Is that right? I tried putting in values to my calc and it looks like the answer should be 1?
What did I do wrong?
Thanks.

 

[Dick]

Check your l'Hopital's rule again. I get 2/pi, not pi/2.

 

[Gib Z]

I also get $2/\pi$, and when I put in x=0.9999 on my calculator, the limit and $e^{2/\pi}$ agree to reasonable accuracy, around 1.89, not 1.