a-function-f-has-a-slant-asymptote-ym-xbm-neq-0-if-lim-_x-rightarrow-inftyfx-m-xb0-andor-lim-_x-ri-6

Question

A function $f$ has a slant asymptote $y=m x+b(m 
eq 0)$ if $\lim _{x ightarrow \infty}[f(x)-(m x+b)]=0$ and/or $\lim _{x ightarrow-\infty}[f(x)-(m x+b)]=0$
In exercises $43-48,$ find the slant asymptote. (Use long division to rewrite the function.) Then, graph the function and its asymptote on the same axes.
$$f(x)=\frac{x^{4}-1}{x^{3}+x}$$

Sinisa Stura · Accepted Answer

Hello. The functions we have in this problem is Russian inflection. Uh, why equal, uh, called every week equals the polynomial X to the fourth month Swerve on extra third class. Mhm. Ah, they want us to find the slam, I think if one exists, So what&#x27;s the deal with Slams s center? It is a line mx plus B that has defined slope. And that&#x27;s what this zero and this line is such that the limit will extends to grow to plus infinity. The difference between the function and the point on the line is becomes very, very small, and the limit to that difference is zero. In other words, if we had a coordinates system and let&#x27;s a slant line and some function and some function whose graph is such that when X becomes greater and greater than the graph off ffx nears, this y equals MX possibility but never crosses it never touches it. Then this is the sliced s entered that off the ground. So if that happens when it tends towards plus affinity or on the other side with extends to minus infinity. So the limit here could have been written and or both sides could be true. Or if either side is true, then the line, um explicitly is smooth Him, given in the problem, is to divide using long division off for a nominal. So with X to the power of three plus six, we&#x27;re going to divide the polynomial X to the fourth minus one. Why have I written it with such forget? Because then we developed polynomial. We want to stress that we have no X to the third no extra second and no term between the ones that we do who have division. We ask ourselves with what do we multiply the leading coefficient off the divisor to obtain the leading coefficient off the dividend? We multiply it x x times X to the power of three 64th six times six is X through the second subtract subtracting means change the ah change the science in ad These to annihilate each other become zero and this becomes one x squared on minus one. So this remainder is now off degree that it&#x27;s more than the degree off the divisor. So we right therefore week as being the Kocian which okay, and then we have this minus X squared plus from This is in fact minus X squared plus one minus X squared plus swim over what we were dividing with X to the power of three. Well, now the trick is thing if we subtract this from a phobic So if we minus sticks is going to be, he&#x27;s going to be mine. This lot here, X great Pass well over it. 36 right? If we take this to be our special with slant line than this limit would be equal to the limits off. Therefore, it minus is when extends to infinity is the same as the limit minus limit when extends the infinite or X squared plus one x to the third plus six. Now, rational functions in limits to infinity are not a huge problem because what we do here is we divide both the numerator on the denominator with the highest with the highest with the highest power of X, which is all right and we obtain limit extends to plus infinity big where divided by external plus well developed externally, its third divided by a third plus X divided by external, which is equal to the limit to infinity. Oh, this can with three of these. So we have one over plus one over the third. These to cancel and leave 1/1. This one on this eggs canceled with these three who will live. So we have one over to wear, right when it He&#x27;s a huge number, then this is yeah, for almost zero from X is huge. Then this is also close to zero, and this is close to zero. So the limit we make approaches plus affinity is zero plus 0/1 0, which is zero, which is zero. And we have. But the limits off f of X 16 is equal to zero. Therefore, this here, this here is the equation off when a simple okay. Now all we need to do is check whether this is true with our graphing device. We use the Christmas here. So it to the fourth month swarm X to the fourth minus world on here We have X to the third person. That right? Excellent. Excellent. Okay, on what we see is as it becomes greater and greater the great the red A grant is the graph of the function black graph is the graph off the line the graph off the function in the line, they become closer and closer. They practically they practically overlap. So from this crap, we see that the line y equals is really this slide. I think that we were off all this time. Okay, Um but here. Right. So as ex moves towards the right far right side that if we becomes closer and closer to why it was a which means that Why? Because X is the ass enter off for weeks. The slip this went on. It&#x27;s the essence of it. But it&#x27;s slight because it has a slope which is not zeros. It&#x27;s not a horizontal essen toe on. It&#x27;s not a vertical, because the vertical has undefined slow. So this is a line with slope different zero such that the function values approach the values off the why coordinates on the line as X moves to the far right or the far left off. Yeah. Breath. So many words. There we have it, folks. Health. Bye.

Sinisa Stura · Answer

Hello. This problem gives us a function F X is equal to except four, um, over X. And the third plus one on the problem wants us to find the slammed, the slant s center. The same essence, which is basically a line that has the light equation. Why can&#x27;t some experts be? It&#x27;s called the slide because this is the slope exist. It&#x27;s zero. So a scientist center is such that the limit when x since two the far right or and the limit when extends to the far left off the graph off the difference off ffx minus this lot MX plus a B. This difference is zero. What this is is the following graphically if we have some sort of a slant line front line. Andi, if this happens in the graph of the function that as we as we move further to the right, the graph seems to get closer and closer and closer and closer to this line, then you know, as X goes to affinity, the difference between the white points off ffx and why equals M X plus B is very, very small. That it is Yeah, it is. The border value is zero. So either when X goes to the far right or the far left if it happens on the other side of the graph, then we say that this here is a slant assistant. The hint given in the in the problem is to divide the numerator with the denominator. Notice that this happens, we&#x27;re going to have a linear function because the because the because the degree off the numerator is by one greater than degree off the, uh, denominator. So when we divided, we&#x27;re going to have a linear Kocian on. We might have some remainder. So let&#x27;s see. We will divide with X to the power of three plus one on the polynomial that we are dividing is X to the power of four plus zero x to the power of three plus zero x the power of two plus zero x plus zero mhm. When we&#x27;re dividing polynomial, this is unnecessary. You don&#x27;t have to write things, but it&#x27;s good to write just so you can keep track of where you are. So how do we do Long division? We take the leading the leading term off the divisor, and we asked, What do we need the most of to get the leading term in the dividend. So x x times X to the third is X to the fourth the next time one is X and we&#x27;re going to right here. Plus subtract means that we can change science and add these two terms violate 0000 We will have minus X plus zero. This will be the remainder because the degree off the remainder is list of the degree off the divisor so we can run That f of X is equal to the questions minus X over X to the power off three plus one. Okay, What this means is that this we will take this to be our m X plus V This lot here because when we subtract f of X minus six, we will be left with limb minus limit off X over X to the power of 3% when x goes to infinity right When we&#x27;re solving we If we subtract, it makes minus x. So if we x minus six is this here? So the limit when x when X goes to infinity will be this limit in these kinds of limits, we divide with with the highest power of X in either the numerator or the denominator. So we will have. This is minus limit extends to plus infinity and we will have one over X squared in the numerator Exa rex third and he will have one plus one over to the power of three, which will when x is a great number, A huge number. This will tend to zero. This will tend to zero. So this limit will be 0/1 0, which is zero on there we have it. Why equals X is the slang? That&#x27;s okay. What&#x27;s left is to check this with our graphic device. We will graph if a vixen graph white six. Um, see, if the function give the graph of the function nears the graph off the line. Fourth over X, the third plus one. So this is going to be thanks thio for over X to the third plus one eyes that what we have Ah, we have the lines. Why equals it now? There is some crossing off lines here, but But what we see, what we want to see is when well X becomes greater and greater. We see that the function. The graph of the function becomes almost They&#x27;re very, very close. You can see that They&#x27;re becoming close here when X is over 33.54 or five and so on. Okay, so what? This graph? What this has told us that we have correctly found are well, our slammed assaulters ready. So once again, when it grows tends towards plus his ability. What happens is that a phobics, the river breath nears the graph off the linear off the line that represents the assistance. And that is the meaning of essence. So it comes very close. Toe never crosses, never touches it. It does cross it here. But in this position, it&#x27;s not that it was not aimed towards infinity x tensile zero and on the other hand, extending to minus infinity, meaning that when x moves to the far left, the same thing happens. It will be comes closer and closer to why you convict. So this line is the client s entered off the given function on there. We have it. Hope it helps, but

Sinisa Stura · Answer

Hello. Look, given the function eggs the power of 31 swarm on X squared My swim. We want to find a slave. A symptom. Ask him to three instructions given by the text are a scholar So slight Essence. That is a line that has, uh, in equation. Why was MX plus be such that the limit went extends to plus infinity off the difference between the function values and the values off the slam s in the palace zero or and the same limits. When X goes to minus infinity off the same thing is equal zero. What this basically means is that if we have a line and this happens, then X goes to infinity. And if off X goes towards the Y off the line that we have here, the hint is the hint is to use long division that is a polynomial on. That is what we&#x27;re going to do. We&#x27;re going to divide extracts of the third with minus one. I&#x27;ll explain why I&#x27;ve written here just in a second X squared, minus one. When we do polynomial division, then we want Thio truth. The the there are missing missing missing terms off the polynomial. So we list all the terms here we have 00 We could have just lifted a zero just as an empty space. So what do our multiply X square to get X to the power of three with x x times X squared is X to the power of three minus x. I write it here. Okay, Andi, I have zero x squared and I have changed the I changed the signs because we will subtract, uh, these two terms from the terms about so these to annihilate each other. 00 zero plus six is just x by itself X once one. Now we have arrived. Uh huh. The remainder, that is, that has a degree list than our divisor is. So we say that if a vic is equal to the cautions, which is a plus x y the swung over X squared minus one. No, If we have a look at this, we will see that the limit went in when X goes to infinity off. This is we&#x27;ll rewrite this as a difference of squares so we can cancel these two. So when X is a huge number, the numerator is one on the denominator is a huge number, which means zero. So if we say that this is X See here this is MX Busby. Then we will have f of X minus X will give us this fraction. We&#x27;ve just found out that the limit off distraction zero. So why equals? It is our slept is our slam s entered. What we need to do now is to check with our graphical calculator. So it&#x27;s the third minus one. Thanks. Third minus one. Um over to the second, minus one on we claim that are slant essence, that is. And is it so? Okay, let&#x27;s change. So, like this to be read. All right, So this is the the graph off f of X and we see as X increases as it moves towards plus affinity. The the graph off a few weeks is ever so closer to the breath or this line. Why minus which basically says that why equals X is the slight s. That&#x27;s what we needed to show. Right? So when extends to infinity than every week, the red right seems to near the graph off the slant s center. Why equals a the limit off a four weeks. Months equals zero. Why? Because F of X minus six is this fraction. Here In this fraction, we have have the era, which means we have answered the questions. Solve the problem. Hope it helps, but

Sinisa Stura · Answer

Hello. We have a function F of X is equal to three. Where the mother swan over X on. We need to find the slam. That&#x27;s something on the way to find the flame s implode is the side s in Turkey&#x27;s wife M X plus B where limit We&#x27;re X. Okay. Mhm infinity. The limit off. Therefore big minus This MX plus day them exposed is equal to zero and or, uh, the limit when it goes to minus infinity off the same thing equals zero So way confined If we can find if we can find a linear function that is along that this the hint given is to use long division. You use long division to rewrite the the ffx, So let&#x27;s divide three square over X minus one over. Since we have a single term in the denominator, this is going to be pretty simple affair. So when we when we canceled the square with the X, we will have three X minus one over X. What this will give us is the expression that we need. This expression here is this expression here. This is our a wants to be okay. So why equals three X is our slang s entered. Why? Well, if we take the limit when a tends towards infinity off F of X, which is three x months, one through it, models. Three years. This this will cancel on this will be one over a huge number. This is zero. So we will have. We will have the the conditions satisfied, and this will be a slide. I think that way will go to our, um, graphing calculator on check whether this is so in a minute. Okay, So we have. And through the route, the red graph is the function of everybody and the green and the green is Let&#x27;s well that the exact to show us greater range of numbers from months. 101 100. I didn&#x27;t help. So we will go from 20 20 but on the why? Well, multiply this by 10. Okay, so once whatever, When we zoom in, we will see that the graph, the really great approaches the green grass has it tends towards minus infinity. And also the river graph approaches. The green grass extends towards possibility. All right, so we will add this working table here. Okay? Sure. This in fact, Yeah. Why equals three X The slammed. So it&#x27;s not a horizontal. It&#x27;s not vertical, but it&#x27;s a slang s entered off the function F of X equals three X spread months. One over there. Hope this helps. Bye.

Sinisa Stura · Answer

Hello. Hello. Hello. We have f of X defined to be three x grade minus juan over X minus one. We need to find the slam slammed s in third this land as symptoms which will be a line with with non zero slope MX plus Be such the limit when x goes either to plus infinity or minus infinity off if off minus that m explosively two x possibly we want the limit off this V zero. If this happens, then what we&#x27;re reading from here is as X moves to the far right than the then the mm. The graph approaches approaches some line, um, expo speak or if it moves to the far left, then the graph will approach some slant line that is some lines with with slope M and interested be. The hint tells us the hint tells us is divide or use long division or rewrite this in a way that will help you find that and then your function. So three X squared minus one, divided by Let&#x27;s let&#x27;s divide. And here we have three x squared minus. That is plus zero. There&#x27;s no excess minus one. We could have done this with synthetic division, but whatever. So what are we divide what we multiplied this with We multiply this three x three x times x three x squared three X times minus one is one is three Change the size. So this this annihilate zero on because becomes three x we lower the risk of polynomial down whether we multiply the leading factor with plus 33 times x is three x three times months one is minus for reid. So change signs. Uh, um, I really and we have a left over to Therefore, uh, a few weeks is equal to, um mm is equal to through X minus One is equal through the quote three X plus three and plus Q over the X minus. One way we&#x27;re dividing this with because if we multiply, if we multiply this predicts one swarm we would have three X plus three that multiply X minus one plus two, which is the way that we check our if our division is correct. So we multiplied divisor with the So this is our M expressivity. Our slant, a flight s center for this function is going to be three x plus for real. We will check this by going to our calculator. Um, so we have f of X is three minus one over x months on their slammed essence. Okay, the red graph is the graph of the function of the green graph is the line that represents the essence that as big as picks tends towards plus infinity. Uh huh. Well, extends towards plus ability. All right, so way have when x goes towards plus infinity, then a few weeks nears. The function nears the graph of three X plus three. Okay, this is it. So when it goes to the far right of the graph off, if comes ever so closer to the graph off three explore three, which is in fact, the interpretation off this equation here and even if it says or or end the same limit. But when X goes to minus infinity So when extends towards violence, affinity when it goes to the left firing off the grass, we also have that effect, gets closer and closer to the graph off three for three, which is the definition off a slam essence. Flint. The solution is straight for three hope the Phillips, but

Sinisa Stura · Answer

hello way are given a function. F of X is equal to the ratio or the fraction off eggs through the third minus two. Great plus one on this is X squared and they want us to find a slang as in third a slam aceptar Ah, once again, what&#x27;s the slant? Aceptar It is aligned. Why equals m x plus v such that the limit when x goes to infinity off the difference between f of X and that line the election is zero or or and the limit when it goes to minus infinity off this lot is equal to zero. So the hint given is that we should use division divide the numerator by the denominator on. From there you will see which limiter function will be would be appropriate to take as slant aesthetic. Okay, so there is only one term in the denominator, so we can say that f of X is equal to x third Liberec squared minus two X squared career plus one over X squared. Some canceling is going to happen here. Canceling is going to happen here and we will have the f of X is equal to my two plus one over X squared now because this will go to zero when it tends to plus or minus infinity then for R m explosively, we take this to be both me and we say that our slant essence there is going to be Why equals X miles? Okay, this is the answer to the question. What is the slight s and third off this function on What we&#x27;re going to do now is check this in a in a You know, graphic calculators such as this was calm. We&#x27;ll say that if they is the ratio off to the third minus who to the second plus form over? Okay, to think. Okay, on we say that why Equals explains to waas are wolf. Our let&#x27;s see was exposed to Okay. What is happening is from the graph we can tell that from this graph we can see and confirmed that this really is true that this is this life essence 30. Why, right? So when it girls tends towards the far right in, then the function f phobics. You see this? What is this color? This is purple. The purple graph is the graph off If and as X moves away from 23456 And so on The we yes, zoom square here and then zooming to see what happens. You see what happens As ex grows and becomes greater and greater, the purple graph becomes closer and closer to the to the graph off. Ah, this line which means that this is in fact this land The slam s entered on clean. All right, so but difference between F Away and X minus two becomes minuscule here, which means that as X grows further to the right, it will be zero. And furthermore, when it goes to minus infinity, we go that way. F of X also nears the nears the breath off the line. Why call six months? So this here is the slam essence off this function on. Then we have the whole solution of the problem. Hope it helps. Bye.

Problem

Find a function whose graph has the given asympto…

Problem 48 Easy Difficulty

Answer

$y=x$

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Robert T. Smith, Roland B. MintonCalculus: Early Transcendental Functions

Grace H.

Anna Marie V.

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Robert T. Smith, Roland B. Minton

Calculus: Early Transcendental Functions