find the limit as x approaches 0 of (1/x) to the x power

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- Aug 2nd 2010, 10:10 PMmarymmlimit
find the limit as x approaches 0 of (1/x) to the x power

- Aug 2nd 2010, 10:19 PMyeKciM
$\displaystyle \displaystyle \lim_{x \to 0} (\frac {1}{x})^x = 1 $

- Aug 2nd 2010, 10:22 PMJskid
Judging by the graph of the function the limit does not exist. But it's bean a while since I've calculated a limit and may be wrong(Doh)

Actually I'm curious how does $\displaystyle \displaystyle \lim_{x \to 0} (\frac {1}{x})^x = 1$ ? The function keeps getting more complex each time a derivitive is taken and since log is undefined at 0 you can't use L'Hopital's rule. - Aug 2nd 2010, 10:54 PMmarymm
we need to take the natural log of y=x ln 1/x

but after that i get confused - Aug 2nd 2010, 10:58 PMmarymm
then the limit

- Aug 2nd 2010, 11:24 PMCaptainBlack
$\displaystyle \ln \left[ \left(\dfrac{1}{x}\right)^x\right]=-x \ln(x) =- \dfrac{\ln(x)}{1/x}$

Now apply L'Hopitals rule:

$\displaystyle \displaystyle \lim_{x \to 0 } \ln \left[ \left(\dfrac{1}{x}\right)^x\right]=\lim_{x \to 0 } \dfrac{1/x}{1/x^2}=\lim_{x \to 0 } x=0$

So the original limit is 1.

CB