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Problem 53 Hard Difficulty

(a) If the symbol [] I denotes the greatest integer function defined in Example 10 , evaluate
(i) $\lim _{x \rightarrow-2^{+}}[x]$
(ii) $\lim _{x \rightarrow-2}[x]$
(iii) $\lim _{x \rightarrow-24}[x]$
(b) If $n$ is an inleger, evaluale
(i) $\lim _{x \rightarrow \infty}[x]$
(iii) $\lim _{x \rightarrow \infty}[x]$
(c) For what values of $a$ dacs lim. $. . .[\mathrm{xl}]$ caist?

Answer

a. $\begin{array}{lll}{\text { i) }-2} & {\text { ii) DNE }} & {\text { iii) }-3}\end{array}$
b. (i) $[x]=n-1$ (ii) $[x]=n$
c. $\lim _{x \rightarrow a}[x]$ exists when $a$ is not an integer.

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Video Transcript

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