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The line $y=m x+b$ is called a slant or oblique linear asymptote of the function whose equation is $y=f(x)$ if either $$\lim _{x \rightarrow \infty}\{f(x)-(m x+b)\}=0$$ Or $$\lim _{x \rightarrow-\infty}\{f(x)-(m x+b)\}=0$$Show that the given linear equation is a slant asymptote of $y=f(x)$ for (a) $f(x)=x^{2} /(x+1), y=x-1 ;$ (b) $f(x)=\left(x^{3}+2 x+3\right) /$$\left(x^{2}+5 x-3\right), y=x-5$ Long division of polynomials may be used to determine the oblique asymptotes of a rational function in which the degree of the numerator exceeds the degree of the denominator by one. For example, performing long division on $$f(x)=\frac{4 x^{4}-8 x^{3}+2 x+5}{2 x^{3}-7 x+9}$$ yields $$f(x)=2 x-4+\frac{14 x^{2}-44 x+5}{2 x^{3}-7 x+9}$$ This implies that the oblique asymptote is $y=2 x-4$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Missouri State University

Oregon 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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(Oblique asymptotes) Let $…

For the given problem, we want to consider the line Y equals mx plus B, which can be a slant or oblique linear assam toe if we have the given function. So looking at, for example, the function that we have, We see that we can end up getting a slight ascent of two X -4. So we'll show an example of this of F of X equals four X. M a forest -8 x cube Plus two x plus five. And this is going to be divided by two X cubed minus seven X plus nine. And we see it has a slant assume toe or an oblique assume two of 2 X -4. So that's the accented that's being approached as X goes to infinity and negative infinity in both directions.

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