Question Video: Finding the Limit of a Function from Its Graph at a Point of Removable Discontinuity If the Limit Exists Mathematics

Video Transcript

Determine the limit as 𝑥 approaches two of 𝑓 of 𝑥 if it exists.

Here, we have been given the graph of 𝑓 of 𝑥, and we’re trying to find the limit of 𝑓 of 𝑥 as 𝑥 tends to two. If we try to find the value of 𝑓 of 𝑥 when 𝑥 is equal to two, we can see that 𝑓 is, in fact, undefined. However, this does not mean we cannot find the limit. We know that the limit as 𝑥 approaches two of 𝑓 of 𝑥 is the value 𝑓 of 𝑥 approaches as 𝑥 tends to two. In order to find this limit, we simply need to consider 𝑓 of 𝑥 around two not specifically at two. Let’s consider the 𝑥-values just to the right and just to the left of 𝑥 is equal to two.

Let’s start by looking at 𝑓 of 𝑥 just to the left of 𝑥 is equal to two. We can see that as 𝑥 gets closer and closer to two from below, the value of 𝑓 of 𝑥 is getting closer and closer to three. And if we consider the 𝑥-values just to the right of 𝑥 is equal to two, then we can see that as 𝑥 gets closer and closer to two from the right, the value of 𝑓 of 𝑥 is decreasing and getting closer and closer to three. Since 𝑓 of 𝑥 is tending to the same value as 𝑥 approaches two from both the left and right and this value is three, we can conclude that the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to three. In this last example, we’ve seen how we still may be able to find the limit of 𝑓 of 𝑥 as 𝑥 approaches a particular 𝑥-value, even if 𝑓 is undefined at that particular 𝑥-value.