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

[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 x`s, 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 x`s in the denominators...
All in all I´m quite lost and have no idea how to find α or(if that's the case) show that α is irrelevant

 

[Nino MMP]

when it comes to f'(0). The only answer I can think of is that α must be greater than 0 so that the function is defined for all x.