True or False: In the given figure, the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to 𝑓 evaluated at one.
In this question, we need to use the given graph of 𝑦 is equal to 𝑓 of 𝑥 to determine whether the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to 𝑓 evaluated at one. To answer this question, let’s start by recalling what we mean by the limit of a function. We can recall that if the values of 𝑓 of 𝑥 approach some finite value of 𝐿 as the values of 𝑥 approach 𝑎 from either side but not necessarily when 𝑥 is equal to 𝑎, then we say that the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. In this case, our value of 𝑎 is one and 𝐿 is 𝑓 evaluated at one. So let’s update our definition with 𝑎 equal to one. This then gives us the following. We could then update the value of 𝐿 to be 𝑓 evaluated at one. However, we can just use the diagram to find 𝑓 evaluated at one.
We can do this by recalling when we graph a function, the 𝑥-coordinate of any point on the curve tells us the input value. And the 𝑦-coordinate of this point tells us the corresponding output for this value of 𝑥. In other words, every point on the curve is of the form 𝑥, 𝑓 of 𝑥. Therefore, to find 𝑓 evaluated at one, we just need to find the 𝑦-coordinate of the point on the curve with 𝑥-value of one. We can do this by using the fact we’re given the vertical line 𝑥 is equal to one. We do need to be careful, however, since there are two points with 𝑥-coordinate one in the diagram.
We need to remember that when there’s a hollow dot in the diagram, the curve is not defined at this point. It’s only defined at the filled point. Therefore, the point with coordinates one, four lies on this curve. So 𝑓 evaluated at one is the 𝑦-coordinate of this point; 𝑓 of one is four. We now want to determine the value of 𝐿. So we need to see what happens to the outputs of our function as the values of 𝑥 approach one from either side. We can do this in exactly the same way by using the diagram. The outputs of the function are the 𝑦-coordinates of the points on the curve.
So let’s start when 𝑥 is greater than one. We can see when 𝑥 is greater than one and our values of 𝑥 are approaching one from the right, the 𝑦-coordinates of these points on the curve are increasing and increasing. They’re getting closer and closer to three. We can see something very similar when our values of 𝑥 approach one from the left. As they get closer and closer to one from the left, the outputs of the function or equivalently the 𝑦-coordinates of the points on the curve are decreasing and decreasing. They’re getting closer and closer to three. And in both cases, this is approaching the same value of three.
Therefore, by following the definition of a limit, we were able to show the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to three. But of course we’ve shown 𝑓 evaluated at one is four. So this limit was not equal to 𝑓 evaluated at one. Hence, it’s false to say about the function given in the question that the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to 𝑓 evaluated at one.