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[-/6 Points]DETAILSGiven thatIim f(x) = 0 lim g(x) = 0 Iim h(x) = 1 Ud 7ca 4F0 lim p(x) = c lim 9(~) = 6, 4d 434evaluate the limits below where possible. (If a limi...

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[-/6 Points]DETAILSGiven thatIim f(x) = 0 lim g(x) = 0 Iim h(x) = 1 Ud 7ca 4F0 lim p(x) = c lim 9(~) = 6, 4d 434evaluate the limits below where possible. (If a limit is indeterminate; enter INDETERMINATE: ) Him [f(x)jekx)Iim (f(x) 1P(x) 17lim (h(x)]r) 73aIim [p(x)]x)Iim [p(x)ja(*)lirp(x)Subinit Answer[4/4 Points]DETAILSPREVIOUS ANSWERStelephone line hangs between two poles 14 m apart in the shape of the catenary Y=10 cosh(x/1o)wher

[-/6 Points] DETAILS Given that Iim f(x) = 0 lim g(x) = 0 Iim h(x) = 1 Ud 7ca 4F0 lim p(x) = c lim 9(~) = 6, 4d 434 evaluate the limits below where possible. (If a limit is indeterminate; enter INDETERMINATE: ) Him [f(x)jekx) Iim (f(x) 1P(x) 17 lim (h(x)]r) 73a Iim [p(x)]x) Iim [p(x)ja(*) lir p(x) Subinit Answer [4/4 Points] DETAILS PREVIOUS ANSWERS telephone line hangs between two poles 14 m apart in the shape of the catenary Y=10 cosh(x/1o) wher



Answers

In Problems $17-38$, find the limit using the properties of limits in Theorem $2 .$ $$ \lim _{y \rightarrow-3} \frac{4 y-6}{3-2 y} $$

So what we can do to this limit is sentenced the limit of a fraction. You can say that this is equal to the limit. As y goes to five of the numerator, six Y -9 divided by the limit. As y goes to five of the denominator. Which is why plus two. What we can do now is we can break up the limit and the numerator and the denominator. And so this is going to be equal to the limit. That's why he goes to five of six, Y minus the limit As why goes to five of 9 divided by the limit. That's why he goes to five of why plus the limit of two. And so now we have a limit of a constant in the numerator and the denominator. Both of those are just going to be equal to their separate constance nine in the numerator and two in the denominator. And that makes sense. Since we can let why go to negative five. And if we're just looking at this constant which is nine, it's never affected by what the y. Value is, it's always going to be equal to nine. So and why it goes to five, it's still equal to nine same one and then why does the five this one's still equal to two? That's why we can say those are equal to those constants. And so this is going to be equal to, I'm actually going to take this um constant out of this limit as well and then multiply it by the limited swaggers to five of just why? So we're gonna have six times the limit. That's why he goes to five of just why minus nine divided by the limit. That's why I used to five of why plus two And the limited swaggers to five of Y is just equal to five. So this is going to be six times five, which is 30 minus nine, Divided by five plus 2, Which is equal to 21 to write it by seven, Which is equal to three.

So we're looking at the limit as y goes negative three of four, Y plus five. Whenever we have a limit of a sum of two terms um we can say that that's equal to the limit of the first term. Just for why? Plus the limit of the second term, Which was just five. And now we have here a constant multiplied by our variable. Why we can take that constant out of this limit. Um Buy property is outlined in theorem to so this is going to be four times the limit. That's why it goes to negative three of why? Plus the limit, This wig is negative three of five. And now we just have a limit of our variable Y in our limit of our constant of a constant. Um And the limit of our variable Y is going to be equal to whatever we're going towards. In this case we're going towards negative three. So this is going to be four times negative three. Plus the limit as y goes to negative three or five is just five. Whenever you have a limit of a constant, it's going to be equal to that constant. And so this is equal to negative 12 plus five, Which is equal to -7.

So here we have the limited back as X approaches one, we have three X squared equals four X minus seven over X minus one. So we're having a deprivation here. We're trying to figure out the slope. We're trying to figure out three things. Return to figure out what f of X is trying to figure out what is, which is the number that were approaching. And we're trying to figure out what are derivative slope is. So I figure out the slope at that point. So the first thing very easy. We're gonna figure out a insulin A is one because we're approaching one. So is equal tour. Next with zero f of X is because we know that I for vexed, has to be equal because we know our deprivation is the limit of X approaches as X approaches a f of X minus f of a over X minus a. We know that after they here has to be a number only enough and because of that and we know that this is equal to one. If we plug in one for the two variables that we have here, we have three times one plus four times one, which is seven. We're subtracting by seven. Thus f of a hiss 7/2 of one. This seven. So based on that logic, f of X equals three x squared plus four X then the last thing we need to do is derived. So we're gonna start by, uh, district factory. So we get three x plus seven times X minus one, divided by X minus What? X minus one Cancels explain this one. And because the limit approaches of except for just one, we get three times 17 bulls 10. So our slope is 10. Our function is three x squared plus four x and A is equal to one, and that's it.

Hello. Hey, we're going expression and were asked to determine the function of, uh, based upon this expression as well as the stuff side number A that we're evaluating at Nebraska calculating woman. So after that, we can have a hint at what our number eight is because we're asked to approach the limit as X goes to 1 may also see it in the denominator that we're gonna be undefined when X equals one. So this told us that our specimen point A is most likely going to be one. Then that brings us to the question What is our potion? And this might be a little tricky because we're looking at it and we're like, Well, that doesn't really look like it's in the form of Ebel's H, uh minus after they. So what? We can dio look at it and try to see if we confected us into two parts and make it a little more sense. But let's evaluate and then return to what we think it could be. So three X squared plus four acts find a Southern can also be written as to factor relation because we have r squared our next to the water and then our constant value, which tells us that it something like plus a or times plus or minus a or something like that. So looking at it, we want a three X even though. And each compartment, because this allows is to get our three x squared when we you are multiplication by part. So what already possible? You know as well we know that we have to have our last part multiply by some factor of seven political seven so really over limited to is something and one because 701 7 So that also told us that one of these is going to be negative story. They're gonna have a negative seven on a positive one or a pilot of seven. Negative, because not feeling a way we can get negative something so we can experience around. But after a little, uh, mixing and matching of these, we see that having seven on the left factory ization is having a negative one on the right. Makes it so that we have are three x squared and then minus three acts plus seven x, which gives us a positive four acts minus seven. So this is equivalent to, um, the same expression to left. And that's how you can do it when you don't really know what you're working with. So if we take this and put it over our X minus one, we see that we have two X minus one. Most canceled. So we're not give in, but women as X approaches one, uh, three X. So when X goes to one, um, this is three times one or seven, which equals Well, that's our limit. But what is our original function? Well, that's gonna be a little trickier, So we know that the derivative can be simplified down to three X plus Southern. So what is the original function? What could the original function be? Well, we can integrate that expression based off of the limit. And that would tell us that perhaps it's three x squared over two plus seven x what I see. But that doesn't really tell us, um, what it could be in terms of, um, having our April's age and our evaluation at a so that's not gonna work. But what could work is looking at this numerator expression and thinking. Well, what happens when we goto one wants. Evaluate, have one. So we have three times one square plus four times one minus. So this is three plus for which goes seven minus seven. Well, that's pretty interesting, because we see that you have a Southern will be evaluated one. And if we're evaluating at one, we have a negative seven in this last part that this tells us that our today is most likely are three x squared. What for Tax? Because if we think about it in terms of, uh, a plus age minus halfway, we have our, um, function of evaluated such that we have, um, there he times one plus h squared, plus four times, one plus age. We can figure that out. But the more important part is what happens when we're evaluating at half of a So we just plug one in. So three times one squared plus four times one. Well, this gives us our of seven. Let's help us that this is pretty good approximation of the potions about all three parts to this. We have the answer to a our first part. So we have a is one. That's our evaluation. What we're evaluating at we have three X squared plus four x, which is our original function. No, we have time, which is our valuation at full of it.


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