# How do you find the limit of ((e^x)-x)^(2/x) as x approaches infinity?

Aug 18, 2016

${e}^{2}$

#### Explanation:

${\left({e}^{x} - x\right)}^{\frac{2}{x}} = {\left({e}^{x} \left(1 - \frac{x}{e} ^ x\right)\right)}^{\frac{2}{x}} = {e}^{2} {\left(1 - \frac{x}{e} ^ x\right)}^{\frac{2}{x}}$

but

${\lim}_{x \to \infty} {\left(1 - \frac{x}{e} ^ x\right)}^{\frac{2}{x}} = 1$ so

${\lim}_{x \to \infty} {\left({e}^{x} - x\right)}^{\frac{2}{x}} = {e}^{2}$