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

Video Transcript

Determine the limit as 𝑥 approaches negative three of 𝑓 of 𝑥.

In this question, we’re asked to determine the value of the limit as 𝑥 approaches negative three of 𝑓 of 𝑥. And to do this, we’re given a graph of our function 𝑓 of 𝑥. And to help us answer this question, let’s start by recalling what we mean by the limit of a function. We say that the limit as 𝑥 approaches 𝑎 of some function 𝑓 of 𝑥 is equal to some finite value of 𝐿 if the output values of our function 𝑓 of 𝑥 are getting closer and closer to 𝐿. Or we can say they’re approaching 𝐿 as the values of 𝑥 approach 𝑎 from both sides.

In other words, to determine the value of the limit of a function, we need to determine what happens to the output values of the function as the input values approach the value of 𝑎. And in our case, we’re trying to determine the limit of 𝑓 of 𝑥 as 𝑥 approaches negative three. And we need to do this by using the graph of our function. Remember, in the graph of a function, the 𝑥-coordinate of a point on the curve tells us the input value and the 𝑦-coordinate tells us the corresponding output value. Therefore, we want to see what happens to the 𝑦-coordinates of points on this curve as our 𝑥-coordinates get closer and closer to negative three. And remember, it’s very important we do this from both sides of negative three.

Let’s start with doing this from the right. That’s values of 𝑥 greater than negative three. Let’s start when 𝑥 is equal to negative one. We can see the output value of our function when 𝑥 is negative one is two. And remember, our input values of 𝑥 need to get closer and closer to negative three from the right. So let’s now try 𝑥 is equal to negative two. We can see the output value of our function is now negative one. And we can keep going in this fashion. As our input values of 𝑥 get closer and closer to negative three from the right, the output values of the function are going to approach the 𝑦-coordinate of this point. That’s negative two.

And we can do exactly the same for values of 𝑥 from the left. We can see that 𝑓 evaluated at negative five is also two. We can also choose values of 𝑥 closer to negative three from the left. We can see when 𝑥 is negative four, our function outputs negative one. And if we continue this process, choosing values of 𝑥 closer and closer to negative three from the left, we can see that our output values of 𝑥 are also going to approach negative two. This means our value of 𝐿 is negative two, since as 𝑥 approaches negative three from either side, the output values of our function 𝑓 of 𝑥 are approaching negative two. And this allows us to say the limit as 𝑥 approaches negative three of 𝑓 of 𝑥 is equal to negative two.

However, there is one interesting thing worth pointing out about this example. We can see when 𝑥 is equal to negative three, our function outputs negative three. This is represented by the solid dot on the graph. And we can see that negative three is not equal to negative two. And this gives us a useful property about limits of functions at points. The value of 𝑓 of 𝑎 does not affect the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥. And this is because, in the definition of our limit, we’re only interested in what happens to the output values of our function as 𝑥 gets closer and closer to 𝑎 from either side. We don’t need to know what happens when 𝑥 is equal to 𝑎.

Therefore, we were able to show the limit as 𝑥 approaches negative three of 𝑓 of 𝑥 is equal to negative two.