Yes, a Graph Can Touch an Asymptote

Date: 06/08/2006 at 09:57:14
From: Janice
Subject: Asymptotes

I am unclear as to whether graphs can cross vertical and horizontal
asymptotes in exceptional instances.  We are putting together a
glossary of math terms and may have to eliminate the part that states
that the graph never touches the asymptotes.  Thank you so much.

Date: 06/08/2006 at 16:49:46
From: Doctor Peterson
Subject: Re: Asymptotes

Hi, Janice.

Yes, a curve may cross an asymptote any number of times!  It is a 
common misconception that it can't EVER touch; the correct idea is 
that although it approaches the asymptote closer and closer as you 
move out along the curve, it never actually reaches the asymptote and 
STAYS there.  That is, the definition of asymptote relates only to the 
behavior of the curve "far out", and it doesn't matter whether it 
ever touches or crosses "close in".  The emphasis should be not 
on "never touching" but on "approaching".

It's unfortunate that many textbook examples are simple, so they 
don't show curves that cross asymptotes.  For example, the following 
gives a (barely) adequate definition, but gives the simplest possible 
example:

    http://mathworld.wolfram.com/Asymptote.html 

A more typical, but still simple, example would be

  y = x/(x^2 + 1)

The graph of that equation has an asymptote at y = 0, but crosses it
at the origin:

                      |       o
                      |  o          o
  <-------------------o-------------------->
         o         o  |
               o      |


How about vertical asymptotes?  Can you see something other than the
definition of an asymptote that would prevent the graph of a function
y = f(x) from crossing its vertical asymptote?  (Hint: why couldn't 
you make a graph like the one above in this case?)

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/