# Determine the exponent(s) α>0 so f'(0) exists

1. Oct 1, 2011

### bhoom

Determine α>0 so that f'(0) exists

$$f_{\alpha }(x)=|x|^{\alpha }sin\left (\frac{1}{x} \right ) , \left [x\neq 0, f_{\alpha }(x)=0 \right ]$$

I derived the function in two cases, one where x<0 and one x>0, and saw that we get x in a denominator three times, As I understand it, it does not matter what α is since f'(0), atm, is undefined. I tried to move around and see if I could get rid of the xs, but I failed.
I also tried to see if I could substitute something(x=e^lnx), but again no luck.

I do not think it is possible to try to find an inverse in some way, since in the end I think that I would get stuck with some x in the denominator.

However, I'm still not sure how some α could fix the xs in the denominators...
All in all I´m quite lost and have no idea how to find α or(if thats the case) show that α is irrelevant