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

Determine lim_(π‘₯ ⟢ 2) 𝑓(π‘₯) if it exists.

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Video Transcript

Determine the limit as π‘₯ approaches two of 𝑓 of π‘₯ if it exists.

Here, we have been given the graph of 𝑓 of π‘₯, and we’re trying to find the limit of 𝑓 of π‘₯ as π‘₯ tends to two. If we try to find the value of 𝑓 of π‘₯ when π‘₯ 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 π‘₯ approaches two of 𝑓 of π‘₯ is the value 𝑓 of π‘₯ approaches as π‘₯ tends to two. In order to find this limit, we simply need to consider 𝑓 of π‘₯ around two not specifically at two. Let’s consider the π‘₯-values just to the right and just to the left of π‘₯ is equal to two.

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

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