Graphing a Function with Asymptotes

Date: 07/05/98 at 01:57:24
From: Nurul Huda Hasan
Subject: Asymptotes of a function

Dear Dr. Math,

I'm having trouble solving problems regarding asymptotes of a function. 
What is an asymptote and how do you solve questions regarding this 
topic? I have tried to solve this one, but I need more guidance to 
solve this question. Thank you for your help.

The question is:

Find the asymptote for this function and draft the graph:

          2x + 1 
   f(x) = ------
          x - 3

Date: 07/07/98 at 12:07:57
From: Doctor Peterson
Subject: Re: Asymptotes of a function

Hi, Nurul. 

Asymptotes can be a little confusing because there are really three 
kinds, each of which needs a different technique. The general 
definition is that an asymptote is a line to which a curve approaches 
as you follow the curve to infinity. The difference is in how you go to 
infinity.

First, a horizontal asymptote is a horizontal line of the form y = c. 
As x goes to infinity, if y approaches c as a limit, then y = c is an 
asymptote. An example would be:

             2
   y = 5 + -----
           x - 3

which approaches:

   y = 5

as x goes to infinity, because (x - 3) goes to infinity and 2/(x-3) 
goes to zero.

Second, a vertical asymptote is a vertical line of the form x = c. You 
approach this case inside-out: as x goes to c, if y goes to infinity 
then x = c is an asymptote. An example is again:

             2
   y = 5 + -----
           x - 3

which approaches infinity as x goes to 3, so it has an asymptote:

   x = 3

Finally, any line y = m * x + b can be an asymptote. Here you can't 
technically just take the limit of the function as x goes to infinity, 
since the limit is infinite, but you can take the limit of the 
difference between your function and a line and see that this goes to 
zero. An example is:

                   2
   y = 2*x + 5 + -----
                 x - 3

which approaches

   y = 2*x + 5

as x goes to infinity.

Now how do you recognize an asymptote? In your case:

       2*x + 1
   y = -------
        x - 3

I would first think informally about what happens when x goes to 
infinity, and where y might go to infinity. When x is very large, 2*x 
will be much bigger than 1, and x will be much bigger than 3, so you 
can ignore the 1 and 3, so (where =~ means approximately equal):

         2*x
    y =~ --- = 2
          x

This suggests that you will have an asymptote y = 2. To prove it more 
carefully, you can use standard limit techniques:

        2*x + 1   2 + 1/x     2
    y = ------- = ------- --> -  as x --> infinity
         x - 3    1 - 3/x     1

When will y be very large? Look in the denominator and ask when 
(x - 3) will be zero. This tells us to check what happens at x = 3:

       2*x + 1   7
   y = ------- = - = infinite
        x - 3    0

so there is another asymptote at x = 3. (If the numerator had also 
been zero, you would have had to look at the limit as x approaches 3.)

Now you have one more thing to do: use this information to graph the 
function. I won't try to draw the graph for you, but what you need to 
do is to draw the asymptote lines y = 2 and x = 3, plot a couple easy 
points, such as where x is 1, 2, and 4 (you want some points near the 
vertical asymptote so you can see how it behaves there), and draw a 
curve that goes through those points and approaches the lines.

I hope that helps you understand asymptotes better. If you have 
trouble with limits, write back and I'll see if I can help you the 
rest of the way.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/