Answer
$\displaystyle\lim_{x\rightarrow 0^-} |x|^{1/x}=\infty$
$\displaystyle\lim_{x\rightarrow 0^+} |x|^{1/x}=0$
The limit does not exist
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{x\rightarrow \pm0} |x|^{1/x}$
Graph the function:
Therefore we get:
$\displaystyle\lim_{x\rightarrow 0^-} |x|^{1/x}=\infty$
$\displaystyle\lim_{x\rightarrow 0^+} |x|^{1/x}=0$
As the left hand limit and the right hand limit are not equal, the limit of the function in 0 does not exist.
