Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

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.$$f(x)=\frac{x^{4}-1}{x^{3}+x}$$

$y=x$

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 6

Overview of Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

Campbell University

Harvey Mudd College

Idaho State University

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

09:08

A function $f$ has a slant…

06:24

05:13

06:22

05:21

01:02

Assume that $f(x)$ has dom…

01:00

01:31

Determine the oblique asym…

01:50

Rational function has an o…

01:48

Graph functions that have …

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's the deal with Slams s center? It is a line mx plus B that has defined slope. And that'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'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'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'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'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'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's the essence of it. But it's slight because it has a slope which is not zeros. It's not a horizontal essen toe on. It'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.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

A function $f$ has a slant asymptote $y=m x+b(m \neq 0)$ if $\lim _{x \right…

Assume that $f(x)$ has domain $[0, \infty)$.Find $\lim _{x \rightarrow x…

Assume that $f(x)$ has domain $[0, \infty)$.Find $\lim f(x)$ if the grap…

Determine the oblique asymptote of the graph of the function.$$f(x)=\fra…

Rational function has an oblique asymptote. Determine the equation of this a…

Graph functions that have slant asymptotes.(a) Show that the line $y=x-2…

06:39

Determine all significant features by hand and sketch a graph.$$f(x)=x^{…

02:57

Numerically estimate the absolute extrema of the given function on the indic…

02:55

Find all critical numbers by hand. If available, use graphing technology to …

02:50

For the cost function in exercise $11,$ show that $C^{\prime}(1000)>\bar{…

03:29

Numerically estimate the limit in (7.1) for $a=\frac{1}{3}$ and compare your…

02:47

03:51

Approximate the area under the curve on the given interval using $n$ rectang…

01:45

Find the production level that minimizes the average cost. $C(x)=0.1 x^{2}+3…

02:21

Use a CAS to determine the range of $x$ 's in exercise 54 for which the…

01:01

Write the given (total) area as an integral or sum of integrals.The area…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.