## Calculus (3rd Edition)

$\displaystyle\lim_{x\rightarrow 0} |x|^x=1$
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: $\displaystyle\lim_{x\rightarrow 0} |x|^x=1$