Question Video: Finding the Limit of a Function from Its Graph at a Point of Jump Discontinuity If the Limit Exists Mathematics

Video Transcript

Determine the limit as 𝑥 approaches five of 𝑓 of 𝑥, if it exists.

We’re given the graph of the function 𝑓 of 𝑥. We need to determine whether the limit as 𝑥 approaches five of 𝑓 of 𝑥 exists. And if it does exist, we need to determine its value. To answer this question, let’s start by recalling the relationship between the limit given to us in the question and the left and right limit. We recall if the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥 is equal to some finite value of 𝐿 and the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥 is also equal to this finite value of 𝐿, then we can say the limit as 𝑥 approaches five of 𝑓 of 𝑥 is equal to 𝐿. And in fact, this relationship works in reverse.

If we know that the limit as 𝑥 approaches five of 𝑓 of 𝑥 is equal to some finite value of 𝐿, then both the left and right limit will also be equal to 𝐿. However, if our left-hand or right-hand limit does not exist, then we say that our limit does not exist. And if our left- and right-hand limit are not equal, then we also say that our limit does not exist. So, one way of determining the limit as 𝑥 approaches five of 𝑓 of 𝑥 is to look at what happens as 𝑥 approaches five from the left and look at what happens as 𝑥 approaches five from the right. Let’s start by evaluating the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥. Since 𝑥 is approaching five from the left, our values of 𝑥 will be less than five.

Let’s see what happens to 𝑓 of 𝑥 as 𝑥 approaches five from the left. Let’s start with 𝑥 is equal to two. We can see from our graph that 𝑓 of two is equal to negative eight. Now, when 𝑥 is equal to three, we can see from our graph that 𝑓 of three is equal to negative three. We can keep going with more values of 𝑥. When 𝑥 is equal to four, we can see from our graph that 𝑓 of four is equal to zero. And in fact, we can keep going with more and more values of 𝑥 getting closer and closer to five from the left. And we can see as our values of 𝑥 get closer and closer, our outputs get closer and closer to one. So as our values of 𝑥 got closer and closer to five from the left, our outputs got closer and closer to one. This is the same as saying the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥 is equal to one.

But remember, we also need to check the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥. We can do this in the same way. This time, our values of 𝑥 will be greater than five. And if we do exactly the same thing, taking inputs of 𝑥 which are getting closer and closer to five from the right, we can see this time something different is happening. This time, it doesn’t matter how close to five we get from the right; our outputs will not get close to one. Instead, it seems to be approaching some small positive number. But what we need to take away from this is the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥 is not equal to one.

So we’ve shown the limit as 𝑥 approached five from the left and the limit as 𝑥 approached five from the right of 𝑓 of 𝑥 were not equal. And remember, when these two limits are not equal, then we say the limit as 𝑥 approaches five of 𝑓 of 𝑥 does not exist. Therefore, by using the graph of 𝑓 of 𝑥, we were able to determine the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥 was equal to one and the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥 was not equal to one. And because these two limits were not equal, we were able to determine that the limit as 𝑥 approaches five of 𝑓 of 𝑥 does not exist.