Question Video: Finding the Value of a Limit from a Graph Mathematics • Higher Education

Video Transcript

The following figure represents the graph of the function 𝑓. What does the graph suggest about the value of the limit as 𝑥 approaches three of 𝑓 of 𝑥?

We’re given a graph of the function 𝑓. And we need to use this to find out information about the limit as 𝑥 approaches three of 𝑓 of 𝑥. And the first thing we’re going to need to recall is what do we mean by the limit as 𝑥 approaches three of 𝑓 of 𝑥. We need to recall that we say the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to some finite value of 𝐿 if the value of 𝑓 of 𝑥 approaches 𝐿 as our values of 𝑥 approach 𝑎 from both sides. And we want to know what the graph tells us about the limit as 𝑥 approaches three of 𝑓 of 𝑥.

So to do this, we’re going to need to ask the question, what happens as 𝑥 approaches three from either side to our outputs 𝑓 of 𝑥? And to do this, we’re given a graph of our function. Remember, the 𝑦-coordinate of the function tells us the output 𝑓 of 𝑥 and 𝑥 is the input of our function, and we want to know what happened to the outputs of our function as our inputs approach three from either side. So let’s start by marking 𝑥 is equal to three onto our diagram, and we can see something very interesting about our function when 𝑥 is equal to three. From the curve, we can see that it is not defined when 𝑥 is equal to three. This is represented by the hollow circle, so we might be tempted to say that this limit does not exist because our function is not defined at this point.

However, let’s take a look at our definition once again. We want to know what happens to our outputs of 𝑓 of 𝑥 as 𝑥 approaches 𝑎. And when we say that 𝑥 is approaching 𝑎, we mean that 𝑥 is getting closer and closer to 𝑎; 𝑥 is never equal to 𝑎. So our function not being defined at this point does not necessarily mean the limit is not defined. So let’s see what happens to our function as 𝑥 approaches three. Let’s start with 𝑥 approaching three from the right. There’s a few different ways of doing this. We could substitute different values of 𝑥 into our equation. Let’s start with 𝑥 is equal to five. When 𝑥 is equal to five, we can see the 𝑦-coordinate of our curve is about 0.5, so 𝑓 of five is approximately 0.5.

Remember, we want to see what happens to 𝑓 of 𝑥 as 𝑥 approaches three. So we need to choose closer and closer values to three, so we’ll do the same thing but this time with 𝑥 equal to four. And when 𝑥 is equal to four, we can see that our curve has a 𝑦-coordinate which is approximately negative 3.5. So 𝑓 of four is approximately equal to negative 3.5. We want to keep choosing closer and closer values to three. And we can see from our curve that as we do this, our values are getting closer and closer to this hollow circle. So as 𝑥 is approaching three from the right, our 𝑦-coordinates are getting closer and closer to negative four. So we would want to choose our value of 𝐿 equal to negative four. However, we need to check the same thing happens as 𝑥 approaches three from the left.

So let’s choose some values of 𝑥 less than three. Let’s start with 𝑥 is equal to one. From our diagram, we can see that when 𝑥 is equal to one, our function 𝑓 of 𝑥 is approximately equal to negative 0.4. We’ll now do the same when 𝑥 is equal to two. This time, when 𝑥 is equal to two, from our diagram, we can see that 𝑓 of two is approximately equal to negative 1.8. And we want to keep choosing closer and closer values of 𝑥 to our value of three. And as we do this, we can see something interesting. Our curve is getting closer and closer to the same hollow circle. As we choose values of 𝑥 closer and closer to three that are less than three, we can see that our outputs are also approaching negative four. This is what we mean when we say both sides. Both of these approach the same value of negative four, so we can set 𝐿 equal to negative four.

Therefore, when we look to the graph of our function of 𝑓, as 𝑥 approached three from both the left and the right, our function 𝑓 of 𝑥 approached negative four in both of these cases. So we were able to conclude that the graph suggests that the limit as 𝑥 approaches three of 𝑓 of 𝑥 would be equal to negative four.