Question Video: Finding the One-Sided Limit of a Function from Its Graph at a Point If the Limit Exists

Video Transcript

Find the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥, if it exists.

We’re given a sketch of the function 𝑓 of 𝑥. We need to use it to determine the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥 if this limit exists. Let’s start by recalling what we mean by this limit. The limit as 𝑥 approaches five from the left of 𝑓 of 𝑥 will be the value that 𝑓 of 𝑥 approaches as 𝑥 tends to five and 𝑥 is less than five. In other words, as our input values of 𝑥 are getting closer and closer to five and 𝑥 approaches five from the left, we want to see what happens to our output values of 𝑓 of 𝑥.

Since we’re looking at the limit as 𝑥 approaches five from the left and we know that our input values of 𝑥 will be on the 𝑥-axis, let’s mark 𝑥 is equal to five. We can see something interesting about our function 𝑓 of 𝑥 at this point. The graph has a hollow circle. This means our function is undefined at this point. In other words, 𝑓 of five does not exist. We might be worried about this, but, remember, we’re only interested in what happens as our values of 𝑥 approach five. This means we don’t need to worry what happens when 𝑥 is equal to five. We’re only interested in what happens as 𝑥 gets closer and closer to five, in this case, from the left.

So let’s see what happens to our output values of 𝑓 of 𝑥 as 𝑥 approaches five from the left. There’s a few different ways of doing this. Let’s start by inputting some values of 𝑥. Let’s start when 𝑥 is equal to two. We can see from the graph when 𝑥 is equal to two, 𝑓 of 𝑥 outputs one, so 𝑓 of two is equal to one. Let’s now see what happens when 𝑥 is equal to three. Either by calculating the gradients of our line or approximating, we can see that 𝑓 of three is approximately equal to two-thirds. And we can do the same at 𝑥 is equal to four. We get 𝑓 of four is approximately equal to one-third.

And we can keep doing this, taking values of 𝑥 getting closer and closer to five. And as we do this, we can see our outputs are getting closer and closer to zero. In other words, we’ve shown as 𝑥 gets closer and closer to five from the left, our outputs 𝑓 of 𝑥 are approaching zero. Therefore, by using this sketch, we were able to show the limit as 𝑥 approaches five from the left of 𝑓 of 𝑥 is equal to zero.