Chapter 1.3, Problem 52ES

The notion of an asymptote can be extended to include curve as well as lines. Specifically, we say that curves $y=f\left(x\right)$ and $y=g\left(x\right)$ are asymptotic as $x\to +\infty$ provided $\underset{x\to -\infty }{\lim }\left[f\left(x\right)-g\left(x\right)\right]=0$ and are asymptotic as $x\to -\infty$ provided $\underset{x\to -\infty }{\lim }\left[f\left(x\right)-g\left(x\right)\right]=0$ In these exercises, determine a simpler function $g\left(x\right)$ such that $y=f\left(x\right)$ is asymptotic to $y=g\left(x\right)$ as $x\to +\infty$ or $x\to -\infty$ . Use a graphing utility to generate the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ and identify all vertical asymptotes.

$f\left(x\right)=\frac{x^3-x+3}{x}$