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Using long division, find the oblique asymptote:$$f(x)=\left(3 x^{4}-2 x^{3}+2 x^{2}-3 x+2\right) /\left(x^{3}-1\right)$$

$$y=3 x-2$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Campbell University

Oregon State University

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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And this problem we are asked to find the oblique asientos using long division of the function you see written here and to use long division. I've gone ahead and written it out in that long division format for us just for the sake of time and what there is to know about this divisor over here, I added these terms that X squared in the X term because they were missing in the usual denominator up here. We want to make sure that we have all of those terms when we're doing long division. So that's just to preface that. Now we can go ahead and start. So we start with this first term X cubed X cubed times what gives us three X to the fourth. Well that's just gonna be three X. But then we can take three x times executed. That is three X to the fourth. Three x times our next two terms of zero are just going to be zero and then three X times that negative one will give us a negative three X. And really what we're doing here is we're subtracting this whole thing from what's directly above it. Going ahead and completing that, you can see that these two are going to cancel out negative two X cubed plus zero is still just a negative two X cubed. So going to have our two X squared and then a negative three X minus a negative three X is going to give us zero. So now this next piece, we're gonna repeat the process. So X cubed again, we're back to this first term. What do we multiply that by to get negative two X cube. Well that's going to be a negative too. And then multiplying this out again. But what we can do right now is actually stop because if we're looking ahead, we can see that these terms up top here this three x minus two. This answer we got that's not going to get. We're not going to simplify that any further. As we do this next step, we're gonna end up finding is just our remainder down here. And when it comes to finding these oblique as sandals, that remainder really isn't important to us. What we want is what we have written up top. So we conclude right away that are oblique. A. Sento is at Y equals three x minus two.

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