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

Determine lim_(π‘₯ β†’ 4) 𝑓(π‘₯), if it exists.

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Video Transcript

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

We’re given a graph of the function 𝑓 of π‘₯. We need to determine whether the limit as π‘₯ approaches four of 𝑓 of π‘₯ exists. And if it does exist, we need to determine its value. There’s a few different ways to determine this value. Usually, the easiest is to look at the left and right limits. Recall if the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is equal to the limit is π‘₯ approaches four from the right of 𝑓 of π‘₯ and both of these are equal to some finite value of 𝐿, then we say that the limit as π‘₯ approaches four of 𝑓 of π‘₯ is also equal to 𝐿.

And it’s worth pointing out this is also true in reverse. So one way of determining the value of this limit and if it exists is to find 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‘₯. If both of these are equal to some finite value of 𝐿, then we’re done. However, if these values are different or one of these or both of these limits don’t exist, we can conclude that our element does not exist. Let’s start with the limit as π‘₯ approaches four from the left.

Remember, our input values of π‘₯ will be on the π‘₯-axis. Since our values of π‘₯ are approaching four from the left, our values of π‘₯ will all be less than four. We want to see what happens to our output values of 𝑓 of π‘₯ as π‘₯ approaches four from the left. Let’s start with π‘₯ is equal to one. We can see that 𝑓 evaluated at one is equal to negative one. If we move closer, using π‘₯ is equal to two, we can see that 𝑓 of two is also equal to negative one. In fact, as we get closer and closer to π‘₯ is equal to four from the left, we can see our output 𝑓 of π‘₯ is always equal to negative one. In other words, the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is equal to negative one.

It’s worth reiterating, even though our function 𝑓 of π‘₯ is equal to negative eight when π‘₯ is equal to four, when we’re evaluating these limits, we’re only interested in what happens near π‘₯ is equal to four. We’re not interested in what happens when π‘₯ is equal to four. We now need to determine the limit as π‘₯ approaches four from the right of 𝑓 of π‘₯. Since π‘₯ is approaching four from the right, our values of π‘₯ will be greater than four. Let’s now see what happens when our values of π‘₯ approach four from the right. Let’s start with π‘₯ is equal to seven. We can see that 𝑓 of seven is equal to one.

When π‘₯ is equal to six, we can see 𝑓 of six is also equal to one. And we can see this pattern continues. As our values of π‘₯ approach four from the right, our output 𝑓 of π‘₯ is always equal to one. So as π‘₯ approached four from the right, our values 𝑓 of π‘₯ approached one. Therefore, we’ve shown them limit as π‘₯ approaches four from the right of 𝑓 of π‘₯ is equal to one. And now, we can see a problem. Our limit as π‘₯ approached four from the left of 𝑓 of π‘₯ was not equal to our limit as π‘₯ approached four from the right of 𝑓 of π‘₯. And we remember if the left-hand and the right-hand limit of 𝑓 of π‘₯ are not equal, then this is another way of saying that our limit as π‘₯ approaches four of 𝑓 of π‘₯ does not exist.

Therefore, by using the graph of 𝑓 of π‘₯, we were able to show the limit as π‘₯ approaches four from the left of 𝑓 of π‘₯ is equal to negative one and the limit as π‘₯ approaches four from the right of π‘₯ is equal to one. And we concluded since the left and right limit were not equal, then the limit as π‘₯ approaches four of 𝑓 of π‘₯ does not exist.

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