Question Video: Finding the Limit of a Function from Its Graph at a Point of Removable Discontinuity If the Limit Exists Mathematics • 12th Grade

Determine lim_(π₯ βΆ 2) π(π₯) if it exists.

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Video Transcript

Determine the limit as π₯ approaches two of π of π₯ if it exists.

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

Letβs start by looking at π 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 π 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 π of π₯ is decreasing and getting closer and closer to three. Since π 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 π of π₯ is equal to three. In this last example, weβve seen how we still may be able to find the limit of π of π₯ as π₯ approaches a particular π₯-value, even if π is undefined at that particular π₯-value.