Question Video: Discussing the Limit of a Function from Its Graph at a Point of Jump Discontinuity

Determine the limit of the function as π‘₯ ⟢ 4, if it exists.

03:26

Video Transcript

Determine the limit of the function as π‘₯ approaches four, if it exists.

We’re given the graph of a function 𝑓 of π‘₯. We need to determine the limit of this function as π‘₯ approaches a four, if this limit exists. To start, since we want to determine the limit of our function as π‘₯ approaches four, let’s look at our graph when π‘₯ is equal to four. We can see there’s a gap in our function at this point. And by looking at the filled-in circle, we know that 𝑓 evaluated at four will be equal to negative three.

But we know when we’re determining the limits of a function as π‘₯ approaches four, we don’t need to worry what happens when π‘₯ is equal to four. We’re more interested in what happens around π‘₯ is equal to four. But this leaves us with a problem. We can see when our values of π‘₯ are less than four, our outputs will all be less than negative three. And we can also see when our values of π‘₯ are greater than four, our outputs will all be greater than one. We know that our outputs can’t approach both negative three and one at the same time.

So to answer this question, we’re going to need to recall the relationship between a limit and the left- and right-hand limit. We recall if the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is equal to some finite value of 𝑙 and the limit as π‘₯ approaches four from the right of 𝑓 of π‘₯ is also equal to 𝑙, then we know the limit as π‘₯ approaches four of 𝑓 of π‘₯ must also be equal to 𝑙. In fact, this relationship is also true in reverse. If we know the limit as π‘₯ approaches four of 𝑓 of π‘₯ is equal to some finite value of 𝑙, then both the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ and the limit as π‘₯ approaches four from the right of 𝑓 of π‘₯ must also be equal to 𝑙.

So, we can determine the limit given to us in the question by looking at what happens as π‘₯ approaches four from the left and what happens as π‘₯ approaches four from the right. Let’s start by looking at what happens as π‘₯ approaches four from the left. This means our inputs would be less than four. From our graph, we can see when we input one into our function, we get an output of negative six. 𝑓 of one is equal to negative six. Similarly, we can see that 𝑓 of two is equal to negative five. And in fact, if we keep going taking points closer and closer to four from the left, we can see that our outputs will be getting closer and closer to negative three.

So as our values of π‘₯ approached four from the left, our outputs 𝑓 of π‘₯ approached negative three. This is the same as saying the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is equal to negative three. We can now do the same as π‘₯ approaches four from the right. From our graph, we can see that 𝑓 evaluated at seven is four. Similarly, 𝑓 evaluated at six is equal to three. And once again, as we take our inputs of π‘₯ closer and closer to four from the left, we can see that our outputs are approaching one. In other words, the limit as π‘₯ approaches four from the right of 𝑓 of π‘₯ is equal to one.

So, we’ve shown that the limit as π‘₯ approached four from the left was not equal to the limit as π‘₯ approached four from the right of 𝑓 of π‘₯. So, we need to recall what happens when these two limits are not equal. If the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is not equal to the limit as π‘₯ approaches four from the right of 𝑓 of π‘₯, then as π‘₯ approaches four, our values of 𝑓 of π‘₯ are not approaching any value. So, we say the limit as π‘₯ approaches four of 𝑓 of π‘₯ does not exist.

Therefore, by using the graph of this function, we were able to show the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is equal to negative three and the limit as π‘₯ approaches four from the right of 𝑓 of π‘₯ is equal to one. And we were able to conclude because these two values were not equal, the limit as π‘₯ approaches four of 𝑓 of π‘₯ does not exist.

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