Video Transcript
Determine the limit as 𝑥 approaches zero from the right of 𝑓 of 𝑥, if it exists.
We’re given a sketch of the function 𝑓 of 𝑥. We need to use this sketch to determine whether the limit as 𝑥 approaches zero from the right of 𝑓 of 𝑥 exists. And if this limit does exist, we need to determine its value. To do this, let’s start by recalling what we mean by the limit as 𝑥 approaches zero from the right of the function 𝑓 of 𝑥.
This is the value that our outputs 𝑓 of 𝑥 approach as our input values of 𝑥 tend to zero and they’re greater than zero. There is one small problem with this definition. Sometimes our output values of 𝑓 of 𝑥 do not approach any one value. For example, our outputs might increase above any value or they might decrease below any value. Another problem might be if our outputs are oscillating around several values. In all of these cases, our outputs are not approaching any one singular value. In all of these cases, we say that this limit does not exist.
However, to check whether our limit does exist, we need to use this method. We need to see what happens to our output values of 𝑓 of 𝑥 as our input values of 𝑥 get closer and closer to zero from the right. This means our inputs will be bigger than zero.
We want to see what happens as our input values of 𝑥 get closer and closer to zero from the right. And there’s a few things that might worry us. For example, our function is not defined when 𝑥 is equal to two. This is represented by the hollow circle in the curve. But remember, our values of 𝑥 are tending to zero. They’re getting closer and closer to zero. So eventually, our inputs will all be smaller than two. So we don’t need to worry that our function is not defined when 𝑥 is equal to two.
With that in mind, let’s pick all of our values of 𝑥 less than two. When we input a value of 𝑥, let’s say 𝑥 is equal to one, we can find the output 𝑓 of 𝑥 by looking at its 𝑦-coordinate. In this case, the 𝑦-coordinate is between negative two and negative one. We can do this with points closer to 𝑥 is equal to zero. And by taking more and more points, we can see that our outputs of 𝑓 of 𝑥 are getting closer and closer to negative seven. In other words, we’ve shown as our input values of 𝑥 are getting closer and closer to zero from the right, our outputs are getting closer and closer to negative seven.
Therefore, not only have we shown that this limit exists, we’ve also shown that it’s equal to negative seven. This means by using the graph, we were able to show the limit as 𝑥 approaches zero from the right of 𝑓 of 𝑥 is equal to negative seven.
