find the limit as x approaches 0 of (1/x) to the x power
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
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.
$\displaystyle \ln \left[ \left(\dfrac{1}{x}\right)^x\right]=-x \ln(x) =- \dfrac{\ln(x)}{1/x}$
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