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### Graphing a Function with Asymptotes

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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
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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/
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Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

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