Derivative using l'hopital's rule

[losin]

let f(x)

exp(-1/x) for x>0, 0 for x<=0

i want to get f'(x) by using l'hopital's rule, but somehow

i'm applying l'hopital's rule again and again and no clear value is coming out.

i know f'(0) is 0, but i cannot prove it

 

[mrbohn1]

So, using the definition of the derivative, you're looking for the limit as x-->0 of exp(-1/x)/x.

If you use L'Hopital on this, you get that it is equal to the limit as x-->0 of -exp(-1/x)/x.

So...?