# What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity?

Mar 1, 2015

A rational function is where there are $x$'s under the fraction bar.

The part under the bar is called the denominator .
This puts limits on the domain of $x$, as the denominator may not work out to be $0$

Simple example: $y = \frac{1}{x}$ domain : $x \ne 0$
This also defines the vertical asymptote $x = 0$, because you can make $x$ as close to $0$ as you want, but never reach it.

It makes a difference whether you move toward the $0$ from the positive side of from the negative (see graph).

We say ${\lim}_{x \to {0}^{+}} y = \infty$ and ${\lim}_{x \to {0}^{-}} y = - \infty$

So there is a discontinuity
graph{1/x [-16.02, 16.01, -8.01, 8.01]}
On the other hand: If we make $x$ larger and larger then $y$ will get smaller and smaller, but never reach $0$. This is the horizontal asymptote $y = 0$

We say ${\lim}_{x \to + \infty} y = 0$ and ${\lim}_{x \to - \infty} y = 0$

Of course ratinal functions are usually more complicated, like:
$y = \frac{2 x - 5}{x + 4}$ or $y = {x}^{2} / \left({x}^{2} - 1\right)$ but the idea is the same

In the latter example there are even two vertical asymptotes, as

${x}^{2} - 1 = \left(x - 1\right) \left(x + 1\right) \to x \ne + 1 \mathmr{and} x \ne - 1$
graph{x^2/(x^2-1) [-22.8, 22.81, -11.4, 11.42]}