Saturday, June 23, 2012


When working with mathematics concepts such as graphing and limits, you will soon encounter lines called asymptotes.  Despite many students' first thought being "that's a weird word!" they are really quite a simple concept to recognize.  In fact, you have probably drawn asymptotes before on your graphs and not even realized that you were using such a fancy mathematical concept.

An asymptote, quite simply, is basically any line (or maybe think of it as a border?) that your graph will approach and get infinitely close to, but never quite reach.  Now, stop and think about this for a minute, and realize how much in common this definition for an asymptotic line has with our definition of a limit which I introduced in a previous post (and then expanded on in the follow up post that discussed limit notation).  Conceptually, they both deal with your given function getting infinitely close to something, and the distance between a point on your graph and either your limit "a" value or asymptote getting infinitesimally smaller but never zero.  I should probably also emphasize this point, because it is not something to be taken for granted either.  Make sure you understand the connection between two things getting closer and closer together without ever touching, and so therefore the distance between the two things gets smaller and smaller without ever reaching zero.  Because if the distance between two points is zero, that implies that they are in the same place or touching, with no space between them to differentiate them.  This may seem like a simple concept, but I want to make sure that everyone understands it.  (For a different perspective, check out Guillermo's post on asymptotes and how they relate to curve sketching!)

So, that is how you could define an asymptote.  But, maybe you'd like to see a picture that more clearly demonstrates asymptotic behaviour of lines.  Let me try to demonstrate briefly with a few graphs:

This first graph is a common one, showing the graph of y = 1/x.

In the graph of y = 1/x, you can see that x never touches the line x = 0.  In this case, you would say that the line x = 0 is called a vertical asymptote.  Similarly for the y values, you can see that y never touches the line y = 0 either, so you would call this a horizontal asymptote.

When drawing graphs, you will often be asked to identify any vertical asymptotes or horizontal asymptotes.  Just make sure you don't mix them up!  :)

Let's look at another one, slightly different this time:

This is related to the first graph.  It is a graph of y = 1/(x-5) +2.

This second example is based on the same function as the first, that is, y = 1/x.  However, it has been horizontally translated to the right 5 units, and vertically translated up 2 units.  In this graph, you can see that the horizontal asymptote is y = 2, and the vertical asymptote is x = 5.  (Even I sometimes get horizontal and vertical asymptotes mixed up, because I instinctively want to refer to the horizontal ones in terms of x, and the vertical ones in terms of y.  However, it's easier to think of it as 'does the line go up and down,' in which case it's a vertical asymptote located somewhere 'side to side' on the x axis.  Similarly, for horizontal asymptotes, they go 'side to side' and are located somewhere along the vertical y axis.)

What we've seen with these two examples, asymptotes are lines that your graph approaches yet will never intersect or touch.  More specifically, they are lines for which your equation is not defined.  In terms of mathematics, one way an equation is undefined is when its denominator equals zero.  So, with that piece of information, it is easier to see why our asymptotes in these equations are what they are.  The equations of these two graphs are undefined, that is, have a denominator of zero, along their asymptotes.

So, that makes reading graphs to determine their asymptotes easier.  However, what about if you are merely given an equation and asked to perform asymptotic analysis on it?  How could you find asymptotes of an equation without seeing it or drawing the graph first?  Well, as I just showed, it's actually quite easy because all you have to do is determine at what points the denominator will equal zero, and you will have then found where the equation is undefined and has an asymptote.

Let's try a brief example to practice asymptotic analysis of the equation.  If you have a graphing calculator and know how to use it, you can plot these out and check.  But otherwise, let's just look at the equations.

Find asymptotes of f(x) = 200 / (x3 - 8).

For each equation you want to evaluate, it's a good idea to consider if there are both horizontal and vertical asymptotes.  To do that, you have to rearrange the expression to solve for either x or for y, and then evaluate each form to determine where the denominator equals zero.  In this example, it is already expressed in terms of x (remember that you can say y = f(x)), so you can simply examine the bottom of this form of the equation, and solve it by equating it to zero:

x3 - 8 = 0
x3 = 8
x = 2

Therefore, this shows that there is a vertical asymptote along x = 2.

To see if there are any asymptotes along the y axis, you need to rearrange the equation to isolate for the other variable.  With some simple algebra, you can rewrite the expression like so:

y = 200 / (x3 - 8)
x3 - 8 = 200 / y
x3 = (200 / y) + 8

When written this way, it is easy to see that the expression will only be undefined when the term 200/y has a zero as the denominator.  And since this is a simple expression, we can easily see that 200/y is undefined when y = 0.

With this quick asymptotic analysis, we have shown that when x = 2, the denominator equates to zero, which then means that f(2) evaluates to 200 / 0, which is undefined.  And if f(x) is undefined when x = 2, we have determined that f(x) has a vertical asymptote at x = 2.  Similarly, we have shown that there is a horizontal asymptote when y = 0, because when we rearranged the original equation to write it in terms of y, we found again that the expression is undefined with the term 200 / 0 when y = 0.

Here is what this function looks like, all graphed out.  Check it out to verify that we identified the correct values for its asymptotes!

Check the graph to see if our asymptotes make sense.  Looks like the check out ok!

So, with that, you now should be familiar enough with asymptotic lines to be able to identify them in your graphs and equations.  Remember that they are similar to limits, in that they are something that something else approaches without ever making contact.  That is a very rough description, but if you can understand that basic concept, you can build on it to be able to apply it more precisely to your mathematics as I've shown in the above graphing examples.  Another very important thought to keep in your mind is that not every equation has an asymptote.  With a little practice and work, you will be able to look at an equation and be able to say exactly where to find its asymptotes, or even declare that the equation has no asymptotes at all.  Being able to identify asymptotes is a core concept in graphing (and is usually one of the first things you want to do when converting an equation to a graph), and will allow you to work through more complicated math questions, such as ones that ask you to "find domain of rational function" or questions that involve oblique asymptotes... ones that slant on an angle!

You will find asymptotic analysis to be very useful in your math studies, so I hope that this post has at least provided a decent introduction for you to this graphing concept.  As always, please +1 the post (below) if you found it useful.  Thanks!

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