
Re: Differentiability
Posted:
Feb 21, 2013 10:17 AM


On Wed, 20 Feb 2013 20:13:24 0800, William Elliot <marsh@panix.com> wrote:
>A problem from >http://at.yorku.ca/cgibin/bbqa?forum=ask_an_analyst&task=list > > Suppose f(x)= e^(1/x^2) for x not equal to 0, and f(0)=0. > > Without using l'hopital's rule, prove f is differentiable at 0 and > that f'(0)=0.
You really _should_ try proving this using l'Hopital and see what happens...
Hint: One way or another, show that
(*) e^x < x (x > 0).
For example, using the power series, or using the fact that
e^x  1 = int_0^x e^t dt
or whatever.
Now what does (*) imply about e^(x) for x > 0?

