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A function $f$ has a slant asymptote $y=m x+b(m \neq 0)$ if $\lim _{x \rightarrow \infty}[f(x)-(m x+b)]=0$ and/or $\lim _{x \rightarrow-\infty}[f(x)-(m x+b)]=0$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.$$f(x)=\frac{x^{4}}{x^{3}+1}$$

$y=x$ is a slant asymptote.

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 6

Overview of Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

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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't some experts be? It's called the slide because this is the slope exist. It'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'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're going to have a linear Kocian on. We might have some remainder. So let'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're dividing polynomial, this is unnecessary. You don't have to write things, but it'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'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'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's okay. What'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're very, very close. You can see that They'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'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

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