Video Transcript
Consider the given figure. Find 𝑓 evaluated at two. Find the limit as 𝑥 approaches two of 𝑓 of 𝑥.
In this question, we’re given a graph of the function 𝑦 is equal to 𝑓 of 𝑥. We need to use it to determine two things. First, we need to find 𝑓 evaluated at two. And next, we need to find the limit as 𝑥 approaches two of 𝑓 of 𝑥.
Let’s start by using this to find 𝑓 evaluated at two. We can do this from the diagram by recalling when we graph a function, the 𝑥-coordinate of a point on the curve tells us the input value and the corresponding 𝑦-coordinate tells us the corresponding output. Since we want to determine 𝑓 evaluated at two, we need to determine the output value of the function when we input a value of two. That will be the 𝑦-coordinate of the point on the curve when 𝑥 is equal to two. And we can find this by adding the line 𝑥 is equal to two onto our diagram.
And when we do this, we can see something interesting. There appears to be two possible answers. However, we need to be careful. Remember, when there’s a hollow dot, the curve is not defined at this point. So the curve is not defined at the lower of these two values because it has a hollow dot. So there’s only one point of intersection between our vertical line and the curve. Therefore, we need to find the 𝑦-coordinate of this point on the curve. Its 𝑦-coordinate is three. And this allows us to find 𝑓 evaluated at two. This point has coordinates two, 𝑓 of two. And the 𝑦-coordinate is three. So 𝑓 of two is equal to three.
Now let’s determine the limit as 𝑥 approaches two of 𝑓 of 𝑥. To do this, we’re going to first need to recall what we mean by the limit as 𝑥 approaches two of 𝑓 of 𝑥. We recall that if the values of our function 𝑓 of 𝑥 approach some finite value of 𝐿 as the values of 𝑥 are approaching 𝑎 from either side but not necessarily when 𝑥 is equal to 𝑎, then we say that the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. And to make this easier, we can start by updating our value of 𝑎 equal to two since we want to determine the limit as 𝑥 approaches two of 𝑓 of 𝑥.
This means we need to determine what happens to the outputs of this function as our values of 𝑥 approach two from either side. And we can do this from the diagram. Remember, the 𝑥-coordinates of points on the curve are the input values and the corresponding 𝑦-coordinates are the corresponding outputs of the function. So we can check what happens to the outputs of our function as 𝑥 approaches two by seeing what happens to our curve as 𝑥 approaches two on the diagram.
But before we do this, there are two things we need to know. We need to see what happens when 𝑥 approaches two from either side. And we’re not interested in what happens when 𝑥 is equal to two. Let’s start with values of 𝑥 greater than two. We can do this in a few ways. For example, we could check what happens when 𝑥 is equal to five. The 𝑦-coordinate of this point is somewhere just above one, so 𝑓 evaluated at five is a little bigger than one. We want to see what happens when 𝑥 approaches two, so we need to take a value of 𝑥 even closer to two, for example, 𝑥 is four. And the 𝑦-coordinate of this point is slightly bigger than the previous one. And we need to keep going in this manner. We need to see what happens to our 𝑦-coordinates as our values of 𝑥 are getting closer and closer to two from the right.
And we can see that our 𝑦-coordinates of these points are getting closer and closer to two. We can do exactly the same thing from the left. We need to see what happens to the 𝑦-coordinates of our points as our values of 𝑥 are getting closer and closer to two from the left. Once again, we can see that these are approaching two. So the values of 𝑓 of 𝑥 are approaching two as 𝑥 approaches two from either side. Therefore, the value of 𝐿 is the 𝑦-coordinate of the hollow dot. It’s two. Therefore, we’ve shown the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to two.
And it’s worth noting we’ve shown something interesting. 𝑓 evaluated at two is equal to three, but the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to two. These two values are not equal, and in general they don’t need to be equal. We can see this in our definition. When we’re taking the limit of a function at a value of 𝑎, we’re not interested in what happens to the function when 𝑥 is equal to 𝑎. There’s no reason to assume 𝑓 of 𝑎 is the same as the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥. And this is an example of this case. Therefore, from the diagram, we were able to show that 𝑓 evaluated at two is three and the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to two.
