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8:40tracs txstate edu1of aak7 HontneneDne [21onVatelantte "CDlocata1uerlnenS7=__ MrEnAneninheTeSECmeah IcltanEenteeemenneeeeane]4 Ceeeednedee...

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8:40tracs txstate edu1of aak7 HontneneDne [21onVatelantte "CDlocata1uerlnenS7=__ MrEnAneninheTeSECmeah IcltanEenteeemenneeeeane]4 Ceeeednedee

8:40 tracs txstate edu 1of aak7 Hontnene Dne [21on Vatelantte "CDlocata 1uerlnen S7=__ MrEnAneninheTeSECmeah IcltanEenteeemenneeeeane] 4 Ceeeednedee



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In Exercises $67-80,$ find the limit of the transcendental function. $$\lim _{x \rightarrow 0} \frac{4\left(e^{2 x}-1\right)}{e^{x}-1}$$

Limit off the sine inverse under X amount of far for extra goes too far on. We need to do this question. Is we blip there far into this ex hair? Therefore we get Nico. Genocide involves under for anybody by far. There we get encouraging decide inverse armed one and saying wasn't one were equally to the Bai Ju.

Hello to everyone. Today we have a problem off limits. So let's suppose that we have the following function that is equals to X minus four, divided by X squared minus three X minus four. And what we want is to know the limit off this function as ex approach to four we're going to do this'll exercise numerically. So this implies that we're not going to obtain an exact value, but a really good approximation. So let's start by constructing the table to get the behavior off the function x near four. That's is the idea off limits. So here we have the 0.4 and we're going to approach from values that are less than four on for values that are greater than than four. And we're going to approach to four in these things. Directions. So to start, let's take values that are less than four. For example, 3.9 and then we take the evaluation at F X. So this is going to be thio 0.204 We're getting closer toe four, and then we obtained, uh, mhm more precision in our calculations than 3.999 This is going to be 0.2000 four. Then we start with the values that are greater than four in X. So this implies 0.1 We can approach as much as we want, but with this is enough to see the behavior of the function. So this have a value of 1.9996 Then we start to getting far from four. We often is your 0.1996 on. Finally, at 4.1, we have think value off 0.196 So what we have here is that if we go from values above for next, we're getting a value near 0.2 on the same happen. If we came from values, uh, less than four, we're getting a on approximation off X near to 0.2. So from here from this table, we get a really good idea of what is happening at four X going to four. So we can say that the limit off F of X at X going to four is approximately zero point on. That's theirs. We can check this graphically off course on here is the plot of the function. So this is Or if x on we can hear, we can go to four, which is a value that is here. So if we take the points near, we're thinking values below. For we're approaching 20.2. And the same happened for values that are greater than four. So we can say that at four. Or if X is equal to 0.2. And that's all the exercise. Thank you.

Okay. So asked Software followed me. So we got worth of our three. That's four times four times four. Well, four times board at 16 and then times that by for and we get six before. Okay, Bren X one. We have big to part two, uh, over me. What that you brute of eighth to dip are up to. And that's equal to what? If you're Cuban of eight to get to work to, which is equal to four and then or this one, he can write that as a positive exponent. That's 1/3 compared to and reading part two is night.

So now we have the following limit, which is the limit as X goes to four for the following function. X divided X plus one minus four divided five on all the expression divided by X minus four. So let's approximate the limit by taking the values near X equals to four. So here we're going to put the values of X and here we're going to put the values off the function. So let's start with 3.9. Then we're going with the values off. 399 then 3999 on then the values that are greater than four. So for 001 for 01 and finally just 4.1. So the values that the function get at this, uh, different values off x R 0.408 here is your 0.0 for 008 here is 0.0 40008 Sorry, eight on then. Here is 0.3999 and here is 0.399 29 on the last value is zero point 039 to. So as you can see, we get a new approximate value off 0.4 because from here you can approach toe points or four. And here is the same thing. So the result in this case is that the limit off this function is an approximate value off 0.0 four and that's it. We can check it this graphically off course. So here is the graph off this function. Well, you can see it completely. This is the behavior of the function. But we're interested in the behavior needs. X equals to four. So is here on. We can check the value here at this point, and, as you can see when X equals to four, then it takes a value of 0.4 in the function.


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