Horizontal and Vertical Asymptotes

Date: 12/5/95 at 3:14:42
From: Anonymous
Subject: question

Dr. Math:

        I am a Math 12 student.  I am confused about the horizontal 
asymptotes and the vertical asymptotes.  Please tell me how to find 
them, and their definition. 

        Thank you very much.

                              Yours truly

                                             David Ju

Date: 1/5/96 at 10:44:35
From: Doctor Ethan
Subject: Re: question

Hey David,

	Great Question.  And I even think that I have an answer for you.

Whenever a function seems to get infinitely close to a line without 
ever crossing it, then that line is a an asymptote.  For example y=1/x

Do you know what it looks like?  I hope so.

It has two asymptotes.  One is vertical and one is horizontal.

That is because 1/x for very large x will get close and closer to 0.
However, it will never be zero.  So the line y=0 is an asymptote.

In the same way, as x gets close to zero, 1/x gets huge.  In fact it 
can get as big as you want.  Formally, when this happens mathematicians 
say that the limit of 1/x as x approaches zero is infinity. So in 
this case the line x=0 is an asymptote.

Now they do get more complex but the general idea is this:
with any x value, where the limit approaches infinity you have a 
vertical asymptote.

Whatever value the function approaches as the x value goes to infinity 
is where the horizontal asymptote is.

Let's do one more example. 

  4
-----
(x-2)

Let's start by letting x get really big.  When x gets big, what 
does the value of the whole thing go toward?

Well 4/(very big number) is going to be very close to zero.
So again we are going to have a horizontal asymptote at zero.

Let's look for vertical asymptotes.

We need to find where the value of the function goes to infinity.  
Do you see why that will be the same places where the bottom goes 
toward zero?

See if you can find those places and they will be the vertical 
asymptotes.

Hope this helps, David.

-Doctor Ethan,  The Geometry/Math Forum