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Using long division, find the oblique asymptotes.$$f(x)=\left(2 x^{2}+x-6\right) /(x-1)$$

$$y=2 x+3$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

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University of Michigan - Ann Arbor

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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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This question is asking us to find the oblique. Ask them to using the method of long division for the function you can see written here F of X is equal to two X squared plus x minus six, divided by x minus one. So let's go ahead and set up our long division where we know that x minus one is divided into our two X squared plus x minus six. First step is determining what we have to multiply X by to get two X squared. That's obviously going to be two X. Then multiplying our two X by X. That's we said we'll give us two X squared two x times a negative one. That we're over here is going to give us a negative two X. But really we're subtracting this whole thing from what we have just above it those two terms. So you can see that two X squared minus a two X. Is going to cancel that off. Positive x minus a negative two X will give us a positive three X. Bringing down our negative six. We'll just repeat the process. So X times what will give us three X. Well that's a positive three. Three times X. Three X three times a negative one is negative three. Again, this gets subtracted. Those are going to cancel out and a negative six minus a negative three is going to give us a negative three. This year is really just our remainder and for the oblique as something that really is not all that important to us. What is important to us is what we have up top right here, there's two X plus three. So we can conclude that our oblique assume toad is Y is equal to two X plus three.

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