# How do you find lim (x+x^-2)/(2x+x^-2) as x->oo using l'Hospital's Rule or otherwise?

Dec 11, 2016

For $x \ne 0$, $\frac{x + \left(\frac{1}{x} ^ 2\right)}{2 x + \left(\frac{1}{x} ^ 2\right)} = \frac{1 + \frac{1}{x} ^ 3}{2 + \frac{1}{x} ^ 3}$

#### Explanation:

${\lim}_{x \rightarrow \infty} \frac{1}{x} ^ 3 = 0$, so

$\lim \left(x \rightarrow \infty\right) \frac{x + \left(\frac{1}{x} ^ 2\right)}{2 x + \left(\frac{1}{x} ^ 2\right)} = \lim \left(x \rightarrow \infty\right) \frac{1 + \frac{1}{x} ^ 3}{2 + \frac{1}{x} ^ 3}$

$= \frac{1 + 0}{2 + 0} = \frac{1}{2}$

The details are shown above, but we can quickly reason as follows.

As $x \rightarrow \infty$ the top goes to $x$ and the bottom to $2 x$, so the ratio goes to $\frac{1}{2}$