# What is the limit of xe^(1/x) - x as x approaches infinity?

Oct 24, 2016

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

#### Explanation:

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

$= {\lim}_{x \to \infty} \frac{{e}^{\frac{1}{x}} - 1}{\frac{1}{x}}$

Direct substitution here produces a $\frac{0}{0}$ indeterminate form. Apply L'Hopital's rule.

$= {\lim}_{x \to \infty} \frac{\frac{d}{\mathrm{dx}} \left({e}^{\frac{1}{x}} - 1\right)}{\frac{d}{\mathrm{dx}} \frac{1}{x}}$

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

$= {\lim}_{x \to \infty} {e}^{\frac{1}{x}}$

$= {e}^{\frac{1}{\infty}}$

$= {e}^{0}$

$= 1$