## Calculus (3rd Edition)

$\displaystyle\lim_{x\rightarrow 0^-} |x|^{1/x}=\infty$ $\displaystyle\lim_{x\rightarrow 0^+} |x|^{1/x}=0$ The limit does not exist
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.