Let [itex]g(x)=e^{-\frac{1}{x}} for x > 0[/itex] and [itex]g(x)=9 for x \le 0[/itex]. I want to prove that derivatives of all orders exist.(adsbygoogle = window.adsbygoogle || []).push({});

Now I know that the only possible problem is at 0. The limit of the difference quotient from the left is obviously going to be 0. The limit from the right is going to be [itex]\frac{1}{x^2}e^{-\frac{1}{x}}[/itex].

But I don't see how this limit exists.

edit: I just checked on maple and it says the limit as x->0 of the above derivative is undefined. What am I doing wrong?

Also, I figure I'm going to use induction to prove f^n, but I can't even get the case where n=1.

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# Derivative question

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