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Topic: Differentiability
Replies: 8   Last Post: Mar 1, 2013 6:22 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Differentiability
Posted: Feb 21, 2013 10:17 AM
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On Wed, 20 Feb 2013 20:13:24 -0800, William Elliot <marsh@panix.com>
wrote:

>A problem from
>http://at.yorku.ca/cgi-bin/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?







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