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Find an equation of the slant asymptote for the graph of the functionTvz +T5 f(z) Vz+5Answer: yCheck...

Question

Find an equation of the slant asymptote for the graph of the functionTvz +T5 f(z) Vz+5Answer: yCheck

Find an equation of the slant asymptote for the graph of the function Tvz +T5 f(z) Vz+5 Answer: y Check



Answers

Find an equation of the slant asymptote. Do not sketch the curve.
$y=\frac{2 x^{3}-5 x^{2}+3 x}{x^{2}-x-2}$

All right. I'm problem. 50 were given the equation of a rational function and were asked to find the equation of any slant Aston totes of that function if there are in so slight passengers air Interesting. So we know already. If we've been looking at horizontal Aston totes, we have a degree three polynomial divided by a degree to polynomial. So I'm gonna write m equals three. So the degree of the numerator and n equals to the degree of his nominator. We should hopefully be familiar with this table. We have a rational function divided by a rational function and then degree divided by degree in. So we're really just looking at the leading terms. And we know that if M is greater than n that there is no horizontal as, um tote. But there is kind of a caveat to this that if m is exactly one mawr than n So the degree of the top is exactly one more than the degree of the bottom. We have a unique circumstance where we can have a science, ask them tote. So if the degree of the top is one more than to give the bottom, we can divide the numerator by the denominator and rewrite it as a linear expression, plus some remainder that will be a lower degree than the denominator so lower degree than that original denominator. And we know that if we have a lower degree on the top and the bottom, that we have a horizontal essentially zero. So as X becomes infinitely large in either the positive or negative direction, this whole thing we have, Ah, horizontal total zero. This whole thing kind of goes to zero. Why equals zero is what it approaches. So as X becomes infinitely large in either direction, we have some linear expression, linear function, plus or minus something that's very close to zero. So that's why, as X goes to infinity and positive negative direction, why equals M X Plus e is what this function will look like. So that's it's a slant, asked him to, because that's what our function will eventually start to look like, as X gets very, very large in either direction. So the remainder should also be a clue that we're going to be dividing here. So because I'm dividing by X squared, I can't do synthetic division. I have to do polynomial long division and be careful that you have to take into account every power of X. So I'm dividing by X squared plus one. But I need to take into account that there is no X term. So I'm gonna write zero X. I need that placeholder or else little throw off my division. Eso I have X squared plus one going into negative X cubed plus three X squared minus X plus one. So polynomial long division works very much the same way that regular long division does. We're just looking, though at the leading term into the leading term on each step. So I have X squared kind of the way I think about it is X squared going into X cubed. But it's really X squared times. What is negative X cubed. So I need to figure out what's missing. Well, I need the negative and I need one more X. So negative X is my first term. What do we do next? On on long division, will we multiply times the entire divisor? So I have negative X times X squared plus zero x plus one and hopefully you'll see why I need that I'm gonna go and distribute. Actually, let's keep you here. This is negative. Execute plus zero X or minus services matter squared and then minus one X. And that is what I'm gonna right here. Negative, X cubed zero X squared minus X. And hopefully you can see why it was so important. Thio, um to have that zero term that I needed thio toe line up the X squared with the X squared. And it's easy to forget to do that. Um, if you don't have that zero x written So now what do we do? Well, with Long Division, we multiply and then we subtract. So I'm going to subtract this whole thing and we get negative X cubed minus X cubed minus negative. Excuse because you have to distribute minus minus becomes plus negative x q plus active zero. Which is why we multiplied by negative X so that that term that first term would cancel out. Now we have minus zero, so three x squared and we have negative X minus negative. X becomes negative X plus x So we have zero X and then I'm gonna bring down the one. Okay, Now I have X squared going into three X squared. Okay, so x times what? I'm sorry. X squared times. What is three x squared? Well, the only thing we're missing for this one is the three. So we have plus three. We multiply three times X squared plus zero x plus one distribute. We get three X squared zero X plus three. And again, it helps to have that zero x term to make sure that we're lining things up correctly. And then again, we subtract. But I'm subtracting the whole things to be careful. Three X squared, minus three squared zero awesome. Zero minus 00 Awesome. One minus three is negative. Two and negative two is a remainder. We know we're done because I know that X squared can't go into a constant. So we know we're going Now I can rewrite my function, which was this, you know, cubic function divided by our cubic, um polynomial divided by quadratic. So we have a negative x cubed plus three x squared minus X plus one divided by X squared plus one which we then we did our long division and we got negative X plus three plus our remainder, which was negative two over what we divided by X squared plus one. So I have, ah, linear expression, plus some remainder. And I know that the remainder is not going to affect my end behavior overly much So. My slant Assam toad is that linear expression. Why equals negative X plus three and that's it.

For probably 49 were given the equation of a rational function and were asked to find the equation of any slant. Assam totes with that function, so scientists and boats were kind of a unique case. We know that because the degree of our numerator is higher than the degree of our denominator that we don't have any horizontal acid totes. But we do notice that our degree is three over to. So a degree of three over degree of two is a special circumstance that when our numerator is one mawr than the denominator. So we know already that there is no horizontal acid joke because our degree on the top is greater than degree on the bottom. But if that degree is one more, so M is one more than n. So the degree of the top is one more than degree of the bottom. We have a special circumstance of a slant Assam tote, so we can divide the numerator by the denominator and rewrite it, um, as a linear term, plus some remainder which will be a lower degree than the denominator. And we know that if we have a lower degree over a higher degree. That's this circumstance that are horizontal, adding to it. Then it zero. So this whole remainder term eventually is X goes to positive or negative infinity. So as it gets really, really large in either the negative or positive direction that this whole term will look like Y equals zero, it will get closer and closer and closer to zero. So as become X becomes infinite in the positive or negative direction, we have this linear expression, plus something that's super close to zero. So that's why y equals MX Plus Constant is our slant Aston toad. So as our function as Ex gets infinitely large in the positive or negative direction, our function looks more and more like that linear expression. So remainder is also a key word that we're gonna be dividing here. So I'm going to divide my numerator by my denominator and because I don't have X plus or minus the number I have X squared. I have to use synthetic are polynomial long division, um, and be careful that you have to take into account every single term, regardless. So the X squared on the bottom. There's there's no X term, and I have to be careful when I set this up that I take that into it account. So I have X squared. I'm gonna write zero X minus One is going into X cubed plus two X squared minus X minus one. Now, with Long Division, we're just looking at the leading term. So for the first, um, we're long division works very much the same way as it does for numbers as it does for polynomial. So I'm just looking, though at the leading terms. So I'm looking at X squared, going into X cubed for this first round. So the way I like to think about this is X squared times What? There's ex cute. I'm looking for what's missing. And in the first round, it's pretty straightforward. It's just x so x squared Times X will give me execute while with long division. We then multiply x times the whole divisor. So I have X Times X squared plus zero x minus one. And I'm hoping on this step it will become obvious why we need that zero terms. So this is X cubed plus zero x squared minus one x someone ray x cubed zero x squared minus X and the significance of that zero is to make sure that my my ex square terms line up. So I needed these terms. Tow line up. Okay, So now what do we do when we divide? Well, the next step is to subtract. So I'm subtracting this whole thing from the original polynomial X cubed minus X cubed zero. Which is why we did it. Tu minus zero is just too. And remember, you have to distribute the negative to every term. So we have negative X minus negative. X becomes negative X plus x Well, it becomes zero x and bring down the minus one. So now, for our next division, we have X squared going into two x squared, so x times what is two X squared times too. So X squared times too. So we have plus two plus two times the whole thing two times x squared plus zero x minus one Distribute. We get two x squared plus zero x minus two just so close to canceling out, we subtract again. So remember, you have to subtract the whole thing. So two X squared minus two x 4 to 0, which is why we did it. Zero minus 00 We have negative one minus negative. Two becomes plus two negative. One plus two is one that is our remainder. So our function ffx we just divided. Execute plus two x squared Cute plus two X squared minus X minus one. We divided that by X squared minus one. And we've got X Plus two with the remainder of one. And it's one over What we divided by so one over X squared minus one. So we have a linear expression, plus some remainder. So this linear expression is our slant Assam tote. So the equation of our explained Assam tote is y equals X plus two, and that's it.

All right. So for problems, 7 47 were given this function. And we need to find this slant Assen took. So before we do that, we needs toe. We need to first check if it actually house so slammed, asking tough. So a function will have a slant ascent of if the degree or the highest exponents of the numerator is a larger than the degree of the denominator. So the highest exponents of the numerator is too, and the highest of exponents of the denominator is one. So the degree of the numerator is larger than the degree of the denominator, which means there is going to be a slight at symptoms. And now, to find that slant hasn't though There are two ways you basically just to divide the numerator by the denominator. And like I said, there are two ways to do the long division or it's intact division. So first we will use the long division. So you take the divisor X minus one. You make a division bracket thus was called and then you put the dividend inside the practice, so X squared plus three X minus threes. Now you do the divine ing So what do you have to multiply? X minus one? Who is so that the leading term will gain X squared? Well, it's just act since next time. Sexist X squared. So excellent access Xcor. That's what you just mentioned. And then x times None of the oneness minus acts. So now he's attract X Square is cancel out. And then there's the three x minus negative X, which is basically plus X 24 acts. And then we bring down the next term, which isn't like the three. And then we continue. So would you have to multiply X minds one so that the leading term ist for X? Well, it's just for so we add plus four to their questions. So four times access for acts four times negative one is 94 so minus four when there was attracts. So the four X's cancel out and negative three of minus negative for or post for us Just one. And just like that, where you found are slant aspecto, which is why is he calls an X Plus four and the remainder than we got this one here really doesn't matter, so we can just completely ignorant and now for a synthetic division. So first we have toe find out like, what's the format of the divisor for us in tech division? Well, the binomial has the divisor Horton since Activision. This written in the form of X minus C. We're seeing a constant number. So our device, or here is X minus one, which means that see, it's gonna be positive one. So with that, we take this number one and we draw like a corner here to separate it from the other numbers. And then we write down the coefficients of each term such that the exponents are decreasing. So it's the excess squared and an axe and then no x. So it's already in the decreasing order, exponents wise. So we can imagine this. That's one times X squared in the front. So it's gonna be one positive three and the nectar three. So one. Sorry. Sorry. So it's game time lagging here. Sorry, three and then number three and then we draw a line and then we bring this first numbers out. So everyone here and then we multiplying this number by the number in the upper left. So one time, someone that salon you bring it back up the line here and then we adds with a number of us, so three plus one is four, and then the cycle repeats so four times one, those four would bring it back up. And then we added, with the number of both so negative three plus Flores one. And just like that, we found the quotient for when you divide this dividend by X minus one. So the question is gonna geo tax plus more with the one that's the remainder. But like I said, it doesn't matter for the sign asking toe. And just like that, we've confirmed that the slant essence of IHS why Sequels and expose for

Yeah, so for this problem we are looking to find a slight nascimento and that's a straight line of the form Y equals mx plus B. Um And that is where the limit X approaches infinity of f of x over X equals A or M in this case. And the limit as X approaches infinity of after vax minus X or mx equals B. Um So first we're gonna find em so looking at this, we want to find the limit of X approaches infinity of x squared plus one over X plus one. That's our acrobatics divided by X. So that could just be multiplied here and we find the limit, we see that the um the degree is the same. So then we just look at the coefficient and this is going to give us one has the limit. Next. We want to find B. So we'll take the limit as x goes to infinity. Um of f of x minus x, that's going to be uh half of x -X, which is just gonna be one X or MX, by the way. Um So what we do is we find a common denominator, we simplify everything and we end up getting that, this is equal to negative one. So now we found our straight line, that's why equals X -1, and that is our final say Nascimento.


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