I was reviewing a problem on an old math contest and it had a problem asking for horizontal, vertical, and slant asymptotes.
The equation was this:
$y=\frac{x^{2}-3x-4}{x^{3}-x^{2}-30x+72}$
Here's a graph:
It said that $y=0$ was a horizontal asymptote. However, the graph shows that it crosses $y=0$, which led to my question: What's the definition of a horizontal asymptote?
Here's something taken from www.freemathhelp.com: "A horizontal asymptote is a y-value on a graph which a function approaches but does not actually reach. Here is a simple graphical example where the graphed function approaches, but never quite reaches, y=0."
The example never said anything about one part of the graph approaching it, but another part intersecting it. That confused me. Thanks in advance.
Actually, it means that a function never reaches its horizontal asymptote when $x\to \infty$. It imposes no restriction for bounded values of $x$.