Answer
$\displaystyle\lim_{x\rightarrow 0} |x|^x=1$
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{x\rightarrow 0} |x|^x$
Compute $|x|^x$ for values of $x$ close to 0:
$|-0.01|^{-0.01}\approx 1.0471285$
$|-0.001|^{-0.001}\approx 1.0069317$
$|-0.0001|^{-0.0001}\approx 1.0009215$
$|-0.00001|^{-0.00001}\approx 1.0001151$
$|0.00001|^{0.00001}\approx 0.99988488$
$|0.0001|^{0.0001}\approx 0.99907939$
$|0.001|^{0.001}\approx 0.99311605$
$|0.01|^{0.01}\approx 0.95499259$
Therefore we got:
$\displaystyle\lim_{x\rightarrow 0} |x|^x=1$
