Indeterminate Form 0^(0) CHALLENGE PROBLEM

Mar 2016
2
0
georgia
I need to find an example of a limit that starts in the indeterminate form 0^0 and then after L'Hopital's Rule, it reduces to anything other than the value of 1. I've been trying for hours now and can't find anything of this form that doesn't reduce to 1. Either I can't see outside the box or it isn't possible.
 

v8archie

Math Team
Dec 2013
7,712
2,682
Colombia
First of all, you don't need l'Hopital's rule.

Secondly, to find the limit of $f(x)^g(x)$ (where $\lim \limits_{x \to 0} f(0)=\lim \limits_{x \to 0} g(0)=0$), you will write it as $\mathrm e^{g(x) \log f(x)}$. Thus we seek functions $f$ and $g$ such that $\lim \limits_{x \to 0} g(x) \log f(x) = k$. Selecting $f(x) = x$ for simplicity, we see that $g(x) = {k \over \log x}$ will do the job, because the $\log x$ terms will cancel for all non-zero $x$.

Solution:
$x^{k \over \log x}$​
 
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