Use the graph shown to find the limit as 𝑥 tends to one from above of 𝑓 of 𝑥.
In this question, we’re asked to find the limit as 𝑥 tends to one from above of a function. Looking at the graph of the function, we can see that 𝑓 of one is six, but this isn’t relevant to finding the limit as 𝑥 tends to one of 𝑓 of 𝑥. When finding the limit, we only look at values of the function around the limit points, not the limit point itself. And in fact, this little plus sign here tells us that we’re only looking at values of 𝑥, which are greater than one.
Looking at our graph, we can see that 𝑓 of two is three, 𝑓 of 1.5 is three, and in fact, 𝑓 of any number which is just greater than one is three. So the value of this limit therefore is three. As 𝑥 tends to one from above; that is, 𝑥 is greater than one, we’re getting closer and closer to one; 𝑓 of 𝑥 remains the same at three. And so this limit is equal to three.
And just as a side note, if we look at the values of 𝑥, which is less than one, we can see that the limit as 𝑥 tends to one from below of 𝑓 of 𝑥 is equal to two. And the fact that the limits from below and the limits from above are not equal tells us that the limit period does not exist.