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Asymptotic Discontinuity: Definition & Concept

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn which type of functions are the most common for producing asymptotic discontinuities. Also in this lesson, see what such a discontinuity looks like when graphed and how you can easily check for their presence.

What is an Asymptotic Discontinuity?

An asymptotic discontinuity is present when you see the graph approaching a point but never touching the point. The same thing is happening on the other side as well. From both sides, it looks like the graph almost touches the point. But because the function never touches the point, it is a discontinuity in the graph.

It is quite interesting to see a function with an asymptotic discontinuity graphed out.

Graphs

Look at this graph, for example.

Example graph.
asymptotic discontinuity

This particular function has an asymptotic discontinuity at x=0. Do you see how the graph approaches x=0 from both sides but the function never touches it? That is the distinguishing mark of such a discontinuity. We have an asymptotic discontinuity at x=0 because that is where our denominator equals 0. Recall that when the denominator equals 0, our function is undefined. An undefined value translates into a discontinuity.

Also, take a look at the function that I just graphed for you.

Functions

The type of functions that produce asymptotic discontinuities are those that have fractions. In math, when you have a fraction of two polynomials, it is called a rational expression. Rational expressions are the most common candidate for looking into the presence of asymptotic discontinuities. Any function, though, that is a fraction at heart can produce asymptotic discontinuities. When I say a 'fraction at heart,' I mean functions that can be rewritten as fractions. For example, the function y=tan (theta) is actually the fraction y=sin (theta)/cos (theta).

The function tan (theta) is actually a fraction.
asymptotic discontinuity

So, if I were to graph y=tan (theta), I would get asymptotic discontinuities where my denominator equals 0. Let's see what that looks like.

Graph of y=tan (x).
asymptotic discontinuity

We see that we have a repetitive pattern of asymptotic discontinuities. Why is that? Because the function cos (theta) is repetitive in nature and reaches zero in a pattern as well. So, if you know that a particular function can be rewritten as a fraction, look to the fraction form when you are working with asymptotic discontinuities.

From this information, we can formulate a method to find asymptotic discontinuities given a particular function.

How to Check for Asymptotic Discontinuities

Because asymptotic discontinuities arise whenever the denominator equals 0, our method is quite simple and involves finding the point where the denominator equals 0. To do that, we simply set the denominator equal to 0 and solve. The solutions will be our asymptotic discontinuities.

Let's try finding some asymptotic discontinuities from the following function.

Does this function have asymptotic discontinuities?
asymptotic discontinuity

At first glance, we see a function that is a fraction. So we can assume that there may be some asymptotic discontinuities present. We don't know what they are just yet. But we will find out shortly.

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