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

# Question Video: Discussing the Limit of a Function from Its Graph at a Point of Jump Discontinuity Mathematics • Second Year of Secondary School

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Determine lim_(π₯ β 4) π(π₯), if it exists.

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### Video Transcript

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

Weβre given a graph of the function π of π₯. We need to determine whether the limit as π₯ approaches four of π of π₯ exists. And if it does exist, we need to determine its value. Thereβ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 π of π₯ is equal to the limit is π₯ approaches four from the right of π of π₯ and both of these are equal to some finite value of πΏ, then we say that the limit as π₯ approaches four of π of π₯ is also equal to πΏ.

And itβ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 π of π₯ and the limit as π₯ approaches four from the right of π of π₯. If both of these are equal to some finite value of πΏ, then weβre done. However, if these values are different or one of these or both of these limits donβt exist, we can conclude that our element does not exist. Letβ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 π of π₯ as π₯ approaches four from the left. Letβs start with π₯ is equal to one. We can see that π evaluated at one is equal to negative one. If we move closer, using π₯ is equal to two, we can see that π 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 π of π₯ is always equal to negative one. In other words, the limit as π₯ approaches four from the left of π of π₯ is equal to negative one.

Itβs worth reiterating, even though our function π of π₯ is equal to negative eight when π₯ is equal to four, when weβre evaluating these limits, weβre only interested in what happens near π₯ is equal to four. Weβ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 π of π₯. Since π₯ is approaching four from the right, our values of π₯ will be greater than four. Letβs now see what happens when our values of π₯ approach four from the right. Letβs start with π₯ is equal to seven. We can see that π of seven is equal to one.

When π₯ is equal to six, we can see π 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 π of π₯ is always equal to one. So as π₯ approached four from the right, our values π of π₯ approached one. Therefore, weβve shown them limit as π₯ approaches four from the right of π of π₯ is equal to one. And now, we can see a problem. Our limit as π₯ approached four from the left of π of π₯ was not equal to our limit as π₯ approached four from the right of π of π₯. And we remember if the left-hand and the right-hand limit of π of π₯ are not equal, then this is another way of saying that our limit as π₯ approaches four of π of π₯ does not exist.

Therefore, by using the graph of π of π₯, we were able to show the limit as π₯ approaches four from the left of π 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 π of π₯ does not exist.

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