# How do you find limits as x approaches infinity?

Nov 7, 2014

Example 1

${\lim}_{x \to \infty} \frac{x - 5 {x}^{3}}{2 {x}^{3} - x + 7}$

by dividing the numerator and the denominator by ${x}^{3}$,

$= {\lim}_{x \to \infty} \frac{\frac{1}{x} ^ 2 - 5}{2 - \frac{1}{x} ^ 2 + \frac{7}{x} ^ 3} = \frac{0 - 5}{2 - 0 + 0} = - \frac{5}{2}$

Example 2

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

since $- \infty \cdot 0$ is an indeterminate form, by rewriting,

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

by l'Hopital's Rule,

$= {\lim}_{x \to - \infty} \frac{1}{- {e}^{- x}} = \frac{1}{- \infty} = 0$

I hope that this was helpful.