WEBVTT
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is problem number 53 of the Stuart Calculus eighth edition
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. Section 2.3 Party of the symbol brackets notes the
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greatest interject function defined an example. 10. Evaluate
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the following limits. So let's take a look and
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refresh our understanding of the greatest interest your function.
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Here we have an example where between say, 10
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and one. The value of the function is zero
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because it is the greatest integer up to that point
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, moving from left to right on the function.
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Once the function reaches the value of one, the
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following integer The function takes on that value of that
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integer until it reaches the next integer to. So
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this seems like a step function where we see that
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the greatest integer is represented in the function value part
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one. The party. What is the limit as
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X approaches negative two from the right of this greatest
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interviewer function Mhm. As we see, we look
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at where negative two is. We follow the function
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from the right towards negative too. And we see
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that we are still along the line of negative two
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. And so this limit that we're approaching is equal
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to negative two. What is the limit as we
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approach negative, too, for this function, well
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, this has to do with the lift, the
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limit from the left and the right, as we
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understand when there is no specification, whether it's from
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the left or the right. Then we determined the
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limit. Based on both of the limits, we
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saw that from the right to the limit as you
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first, negative two is negative. Two. As
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you approach negative two from the left, we see
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that limit is equal to negative three. So because
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the limit from the left and the right do not
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agree at negative two, we say that the limit
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does not exist. Part three. The limited expertise
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. Negative 2.4. For this integer function, the
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2.4 falls somewhere between negative two and three. If
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we go down to the function, notice that it
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is a constant value, so negative 2.4. Here
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we have to understand where the limit is coming from
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the left and the right notice that no matter where
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you are, between these two indicators negative three negative
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two. The limit is always going to be equal
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to negative three because the function stays constant from both
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the left and the right towards any point between those
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two integers. So the limit here does exist,
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and it's equal to negative three. Puppy. If
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N is an integer, evaluate the limit as X
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approaches end from the left. Let's take a look
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here. Let's pick a value N for an example
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here and is one, Let's say, And if
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we're approaching one from the left, we see that
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the limit of that function in zero. Let's take
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the next value to as we approach and is equal
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to two. From the left, we see that
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the value of function as one. So we see
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that as you approach this function, you approach a
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value event from the left, you get a value
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that's one less than the value of N. And
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so we conclude that the limit is always going to
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give you one less than the value of N as
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the limit from the other direction approaching end from the
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right, let's say we choose N is equal to
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one from the right, we see that the limit
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is equal to one. Let's say we chose and
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is equal to negative one. If we approach and
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equals negative one from the right. We are at
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negative one, and this is characteristic of the entire
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function. That the limit, as you approach end
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from the right for this function is equal to the
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value and itself. Now, for part C for
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what values of aid is the limit, as X
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approaches in of the greatest interest function exists well,
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we noticed that the limit did not exist when X
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equals negative two. Here we see that every time
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there's a jump discontinuity in the function, the limit
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does not exist, and we see that the jump
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occurs at every time that there's a new integer negative
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to negative 101 and so on. So the limit
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as the expertise eight exists for all values. Besides
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a is an integer, as we see here,
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when a is an integer imagining and is an integral
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here. Notice that the limit from the left is
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different from the limit from the right, always but
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as we saw for any value between the two integers
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, such as in the example, Part three of
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a, you are always going to get a limit
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that exists because the value is constant. And when
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you approach the any point from the left and the
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right, you will always get that same constant value
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. So we say that the values no way that
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satisfied this in it are all numbers all real except
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a equals unreal except and teachers, and that concludes
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our solution.