Asymptote for (x-4)/x^2

Date: 04/10/2002 at 00:57:49
From: Shunt Jarian
Subject: Asymptote for (x-4)/x^2

My math book says that for the function (x-4)/x^2 there is a 
horizontal asymptote at y=0. However, at x=4 y=0 the graph of the 
function does intersect the line that is supposed to be the asymptote. 
How can this be if by definition the asymptote is a line that the 
function gets close to but does not touch?

Thank you,
Shunt J

Date: 04/15/2002 at 05:59:42
From: Doctor Floor
Subject: Re: Asymptote for (x-4)/x^2

Hi, Shunt,

Thanks for your question. A very good question it is!

Your definition of an asymptote is the way we talk about it, but it 
is not really precise. That is, you would have to say "if you start 
far enough away, then the asymptote is a line that the function gets 
infinitely close to, but does not touch."

In your case, it is clear that we should not bother about a point of 
intersection at (4,0), or even (40000000000000,0). If only beyond 
that point there are no more points of intersection.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/