WEBVTT
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Determine the limit as π₯ approaches five of π of π₯, if it exists.
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Weβre given the graph of the function π of π₯.
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We need to determine whether the limit as π₯ approaches five of π of π₯ exists.
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And if it does exist, we need to determine its value.
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To answer this question, letβs start by recalling the relationship between the limit given to us in the question and the left and right limit.
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We recall if the limit as π₯ approaches five from the left of π of π₯ is equal to some finite value of πΏ and the limit as π₯ approaches five from the right of π of π₯ is also equal to this finite value of πΏ, then we can say the limit as π₯ approaches five of π of π₯ is equal to πΏ.
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And in fact, this relationship works in reverse.
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If we know that the limit as π₯ approaches five of π of π₯ is equal to some finite value of πΏ, then both the left and right limit will also be equal to πΏ.
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However, if our left-hand or right-hand limit does not exist, then we say that our limit does not exist.
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And if our left- and right-hand limit are not equal, then we also say that our limit does not exist.
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So, one way of determining the limit as π₯ approaches five of π of π₯ is to look at what happens as π₯ approaches five from the left and look at what happens as π₯ approaches five from the right.
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Letβs start by evaluating the limit as π₯ approaches five from the left of π of π₯.
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Since π₯ is approaching five from the left, our values of π₯ will be less than five.
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Letβs see what happens to π of π₯ as π₯ approaches five from the left.
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Letβs start with π₯ is equal to two.
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We can see from our graph that π of two is equal to negative eight.
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Now, when π₯ is equal to three, we can see from our graph that π of three is equal to negative three.
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We can keep going with more values of π₯.
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When π₯ is equal to four, we can see from our graph that π of four is equal to zero.
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And in fact, we can keep going with more and more values of π₯ getting closer and closer to five from the left.
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And we can see as our values of π₯ get closer and closer, our outputs get closer and closer to one.
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So as our values of π₯ got closer and closer to five from the left, our outputs got closer and closer to one.
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This is the same as saying the limit as π₯ approaches five from the left of π of π₯ is equal to one.
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But remember, we also need to check the limit as π₯ approaches five from the right of π of π₯.
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We can do this in the same way.
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This time, our values of π₯ will be greater than five.
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And if we do exactly the same thing, taking inputs of π₯ which are getting closer and closer to five from the right, we can see this time something different is happening.
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This time, it doesnβt matter how close to five we get from the right; our outputs will not get close to one.
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Instead, it seems to be approaching some small positive number.
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But what we need to take away from this is the limit as π₯ approaches five from the right of π of π₯ is not equal to one.
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So weβve shown the limit as π₯ approached five from the left and the limit as π₯ approached five from the right of π of π₯ were not equal.
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And remember, when these two limits are not equal, then we say the limit as π₯ approaches five of π of π₯ does not exist.
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Therefore, by using the graph of π of π₯, we were able to determine the limit as π₯ approaches five from the left of π of π₯ was equal to one and the limit as π₯ approaches five from the right of π of π₯ was not equal to one.
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And because these two limits were not equal, we were able to determine that the limit as π₯ approaches five of π of π₯ does not exist.