Question Video: Discussing the Existence of the Limit of a Piecewise-Defined Function at a Certain Point Mathematics

Video Transcript

Discuss the existence of the limit as 𝑥 approaches four of 𝑓 of 𝑥 given 𝑓 of 𝑥 is equal to six if 𝑥 is less than four and 𝑓 of 𝑥 is equal to two if 𝑥 is greater than four.

In this question, we’re given a piecewise-defined function 𝑓 of 𝑥 and we’re asked to discuss the existence of the limit as 𝑥 approaches four of 𝑓 of 𝑥. And before we recall what we mean by the existence of a limit, we should start by taking a look at our function around the value of 𝑥 is equal to four. And there’s a few different reasons for this. First, there’s lots of different ways a limit cannot exist. So, instead of recalling all the different ways a limit might not exist, we should start by taking a look at the outputs of our function near 𝑥 is equal to four. We can then see what problems arise when we try and find a value for this limit.

Another reason is our function is a piecewise-defined function. This means it’s several different subfunctions defined over several different subdomains. In particular, one problem that can occur is when our limit point is at the endpoints of the subdomains of our subfunction. This means that our function is defined differently to the left and to the right of our limit point. We can see this is what’s happening in this case. Four is the endpoint of the subdomains of 𝑓 of 𝑥.

And at this point, we can start to notice something interesting. When our input values of 𝑥 are less than four, our function 𝑓 of 𝑥 is outputting a constant value of six. However, when our input values of 𝑥 are greater than four, our function is outputting a constant value of two. So it appears our function has different left and right limits as 𝑥 approaches four. We can use this to discuss the existence of this limit.

We recall that we say the limit as 𝑥 approaches 𝑎 of a function 𝑓 of 𝑥 exists and is equal to a finite value of 𝐿 if the limit as 𝑥 approaches 𝑎 from the right of 𝑓 of 𝑥 and the limit as 𝑥 approaches 𝑎 from the left of 𝑓 of 𝑥 both exist and are both equal to 𝐿. In other words, for the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 to exist, we need to check three things. First, we need to check that the limit as 𝑥 approaches 𝑎 from the right of 𝑓 of 𝑥 exists. Second, we need to check that the limit as 𝑥 approaches 𝑎 from the left of 𝑓 of 𝑥 exists. Third, we need to check that both of these are equal to the same value of 𝐿.

But we can see this is not what’s happening in this case. For example, we could do this using a diagram. However, we can actually do this directly by using limits. Let’s start by finding the limit as 𝑥 approaches four from the left of 𝑓 of 𝑥. Since we’re taking the limit as 𝑥 approaches four from the left, our values of 𝑥 are always less than four. And in this case, our function 𝑓 of 𝑥 is equal to a constant value of six. Therefore, this limit is just equal to the limit as 𝑥 approaches four from the left of six. But six is a constant, so its value doesn’t change as our value of 𝑥 changes. So this limit evaluates to give us six.

We can do exactly the same to determine the limit as 𝑥 approaches four from the right of 𝑓 of 𝑥. This time, our input values of 𝑥 are greater than four. So our function 𝑓 of 𝑥 is equal to the constant value of two. Therefore, this limit is just equal to the limit as 𝑥 approaches four from the right of two. And two is a constant, so this limit just evaluates to give us two. So the left limit was equal to six and the right limit was equal to two. These are not equal. So we can say that the limit as 𝑥 approaches four of 𝑓 of 𝑥 does not exist.

And in particular we can use the notation 𝑓 evaluated at four from the left to represent the limit as 𝑥 approaches four from the left of 𝑓 of 𝑥 and 𝑓 evaluated at four from the right to represent the limit as 𝑥 approaches four from the right of 𝑓 of 𝑥. We can then conclude the limit does not exist as 𝑓 evaluated at four from the left is not equal to 𝑓 evaluated at four from the right.

It’s worth noting we can also check this answer by considering a graph of 𝑓 of 𝑥. 𝑓 of 𝑥 is a constant value of six when 𝑥 is less than four and a constant value of two when 𝑥 is greater than four. So it’s the horizontal line 𝑦 is equal to six when 𝑥 is less than four. And it’s the horizontal line 𝑦 is equal to two when 𝑥 is greater than four. This gives us the following. We can then see the left and right limit from the diagram. As 𝑥 approaches four from the left, our outputs are a constant value of six. So the left limit is six. However, as our values of 𝑥 approach four from the right, our outputs are a constant value of two. So our limit as 𝑥 approaches four from the right is two. These are not equal. So we can also conclude that the limit as 𝑥 approaches four of 𝑓 of 𝑥 does not exist because 𝑓 evaluated at four from the left is not equal to 𝑓 evaluated at four from the right.