In this explainer, we will learn how to evaluate one-sided limits graphically and algebraically.
Suppose that the function is defined near , but not necessarily at itself.
The easiest way to get such a function is to start with another function, say , which is defined everywhere, and then define another function by setting for every .
Notice that and are different functions, because they do not have the same domain. For example, taking and , we have the function whose graph is shown in the figure.
The hollow circles at and indicate that is not in the domain of the function . Of course, the graph of is just the entire line with the point included, and .
The idea of “the limit of the values as approaches from the left” is that we are looking for a number so that gets closer and closer to as moves through values in the domain of , but only those values that satisfy .
For example, we could look at the values along the -axis, which are approaching , one decimal at a time, and follow along the -axis, through values
We see that the limiting value appears to be just . This is written as and read as “the one-sided limit of as tends to from the left (or below) is .”
This fact does not require a listing of values as we did above: it is fairly clear from the graph, as is the fact that the limit as we approach from the right (above) is also , which we write as
A more advanced—and common—example of a function defined everywhere except at is one given by a piecewise definition: which has the following graph.
Now, just from the graph, the right-sided limit is the same as before, and the left-sided limit can only be the given constant value
This second case could also have been a piecewise-defined function that was defined even at . For example, all of which have different values , , and . But all have the same behavior as regarding one-sided limits: and similarly for the left-sided limits. This is because those limits are only interested in how the functions behave away from , where all four functions are exactly the same.
Example 1: Finding One-Sided Limits of a Function from Its Graph
Use the graph shown to find .
For the graph, the function is defined as when . Therefore, the same constant.
Be sure not to get confused with the position of the “” and especially the “” signs in the problem. The ones appearing as superscripts on the right side indicate whether we are looking at right-sided or left-sided limits, respectively.
Example 2: Finding One-Sided Limits of a Function from Its Graph
From the graph, we see that, in spite of the value , the limiting value as from the left is 2:
When does a one-sided limit not exist? A simple example is provided by with its domain .
As we see from its graph, it makes sense to say the values become infinitely large positive as , and we express this (and similarly, what happens as ) by writing
In other words, there is the notion of “infinite limits.”
Example 3: Finding One-Sided Limits of a Function from Its Graph
Looking at the vertical asymptote at , we see that as approaches this value from the right-hand side, becomes increasingly large and negative, so
It is worth noting the following example, where really does not exist, while does.
The function “oscillates” more and more rapidly between 1 and as from the right-side, so that it never settles on a limit. As from the left-side, the graph indicates that the values are approaching 0.25. Finally, the actual value of this function at is 1.1. So
In fact, the description of this function is
In order to understand the “oscillation,” consider at what values we have . Since it follows that and therefore, looking near , but only to the right, we have which gives a sequence of points on the number line that approach while the difference between successive terms gets smaller and smaller. These are the numbers where the graph above crosses the -axis.