Limits of Piecewise Functions  
 
Introduction  
This lesson page will inform you how to find various limits of piecewise functions. Here are the sections within this page: Before getting started, please familiarize yourself with our previous two lessons regarding limits. These lessons are essential for understanding the following sections within this lesson page.
esson: Limits of the Extreme Likewise, an essential part of understanding the limits of piecewise functions is to first understand piecewise functions. Review this lesson to learn more about piecewise functions.
esson: Piecewise Functions

When we determine a limit of a function, we attempt to see if there is a trend. Without actually evaluating the function at a specific xvalue, we look to see what is happening to the yvalues as we get closer to a certain xvalue. If we were given the function f(x) that has been graphed below, we can determine the limit of the function as we approaches the xvalue 1. If we are left of the xvalue 1 and we move to the right, the yvalues get larger. As we approach the xvalue 1, the yvalues get closer to 1. View the graph to confirm this fact.
Likewise, if we are right of the xvalue 1 and we move to the left, we will notice how the yvalues get smaller. As we approach the xvalue 1, the yvalues get closer to 1. No matter if we approach the xvalue 1 from the left or the right, the yvalue approaches 1.

Within the last section (see Evaluating Limits), we performed two limits. The first limit can be written like so.
It is pronounced as 'the limit of f(x) as x approaches 1 from the left. [The negative sign that is hovering on the 1 that looks like it is in the exponent position is the clue that we are being asked to approach the xvalue 1 from the left side of 1.] Similarly, we can write the limit as we approach the xvalue 1 from the right side, like so.
Again, the plus sign that is hovering in the exponent position mean that we are being asked to approach the xvalue 1 but this time from the right side of 1.
 
There are two cases that can happen when dealing with limits. Case 1: The Limit Exists For some avalue, if...
...then...
Case 2: The Limit Does Not Exist For some avalue, if...
...then...
DNE is shorthand for 'does not exist.' We already saw an example of case 1 (see the example within Limit Notation). So, we will examine a situation for case 2. The graph below is of g(x). We will use this graph of g(x) to determine the following limit.
To determine this limit, we have to first consider both of these limits separately.
First, we will consider this limit.
To calculate this limit, we have to approach the xvalue 3 from the left side. We have to observe the yvalues and see where they are headed as we move to the right, closer to the xvalue 3. As we move closer and closer to the xvalue 3, we can see that the yvalues remain constant at 3. Now let's calculate the other limit.
We are being asked to approach the xvalue 3, but from the right side of 3. As we move left along the curve, we eventually slide downward on the left side of the semicircle. As we get closer to the xvalue 3, we can see the yvalue is moving toward 0. Notice how the two limits are not equal.
Since the two limits are not equal, we say...
...which means the limit 'does not exist.'  
Try this interactive quiz, which are related to the sections above.  
Try these lessons, which are related to the sections above.
esson: Functions 