Video: Finding the Limit of a Function from Its Graph If the Limit Exists

Video Transcript

Using the graph shown, determine the limit as 𝑥 tends to three of 𝑓 of 𝑥.

Here, we have the graph of 𝑓 of 𝑥. And we’ve been asked to find the limit as 𝑥 tends to three. We can see that at 𝑥 is equal to three, 𝑓 of 𝑥 is defined to be negative five. However, when we are finding the limit of a function at a particular point, the value of that function at that point does not matter. What matters is what’s happening to the function around that point. This is because the limit as 𝑥 approaches three of 𝑓 of 𝑥 is defined to be the value 𝑓 of 𝑥 approaches as 𝑥 tends to three. Let’s consider what’s happening to 𝑓 of 𝑥 to the left and to the right of 𝑥 is equal to three.

If we consider 𝑓 of 𝑥 to the left of 𝑥 is equal to three, we can see that 𝑓 of 𝑥 is increasing and getting closer and closer to the value of two, as 𝑥 is getting closer and closer to three. And as 𝑥 approaches three from the right, 𝑓 of 𝑥 is again increasing. And it is also getting closer and closer to the value of two. Now, this tells us what we need to know about 𝑓 of 𝑥 as 𝑥 tends to three, since from both the left and right, the value of 𝑓 of 𝑥 approaches two as 𝑥 approaches three. And so, even though the value of 𝑓 of three is equal to negative five, the limit as 𝑥 approaches three of 𝑓 of 𝑥 is equal to negative two, which is our solution to this question. In this previous example, we’ve seen how even though a function may be defined at a different point at a particular 𝑥-value, the limit as 𝑥 approaches that particular 𝑥-value of 𝑓 of 𝑥 may be different to the value of 𝑓 of 𝑥 at that point.