Question Video: Finding the One-Sided Limit of a Function from Its Graph Mathematics

Video Transcript

Use the graph below to find the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥.

We’re given a graph of the function 𝑓 of 𝑥. We need to use this graph to determine the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥. To do this, let’s start by recalling what we mean by the limit as 𝑥 approaches five from the right of a function 𝑓 of 𝑥. This is equal to the value 𝑓 of 𝑥 approaches as 𝑥 tends to five where 𝑥 must be greater than five. In other words, as our input values of 𝑥 approach the value of five from the right, we want to know what happens to our output values of 𝑓 of 𝑥.

We can do this by using our graph. We know that our input values of 𝑥 will be on the 𝑥-axis. So let’s mark 𝑥 is equal to five. We might be worried from our sketch that when we input 𝑥 is equal to five, we can see that 𝑓 of 𝑥 is undefined. This is represented by the hollow circle. But remember, we’re only seeing what happens to our outputs as 𝑥 approaches five. And in this case, 𝑥 must be greater than five. 𝑥 is never equal to five.

So let’s see what happens to our outputs of 𝑓 of 𝑥 as 𝑥 approaches five from the right. First, we can see that our function 𝑓 of 𝑥 is not defined when 𝑥 is greater than eight. So we should pick inputs smaller than this. Let’s start with 𝑥 is equal to seven. From our diagram, we can find the output of 𝑓 of 𝑥 when 𝑥 is equal to seven. We get 𝑓 of seven is equal to negative six. We can do the same when 𝑥 is equal to six. We see that 𝑓 of six is also equal to negative six.

In fact, we can see exactly the same thing happens as our values of 𝑥 approach five from the right. They always output negative six. And if our outputs are remaining constant at negative six, we can also say they’re approaching negative six. Therefore, we’ve shown as 𝑥 approaches five from the right, 𝑓 of 𝑥 approaches negative six. Or, in other words, the limit as 𝑥 approaches five from the right of 𝑓 of 𝑥 is equal to negative six.