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In the previous exercise, do $m$ and $n$ have to be integers, that is, must $f$ be a rational function?Suppose the rational function $r(x)=q(x)+l(x)$ where $l(x)$ approaches 0 as $x$ approaches $+\infty$ or $-\infty,$ then $r(x) \rightarrow q(x)$ as $x$ approaches $+\infty$ or $-\infty,$ or we say $r(x)$ is asymptotic to $q(x) .$ For example, $$r(x)=\frac{2 x^{2}+6 x+8}{x+1}=2 x+4+\frac{4}{x+1}$$ as $x$ approaches $+\infty$ or $-\infty, 4 /(x+1) \rightarrow 0,$ so $r(x) \rightarrow 2 x+4 .$ Therefore $y=2 x+4$ is an asymptote for $r(x) .$ A sketch is given is Figure $34 .$ Note how the graph approaches the line $y=2 x+4$ as $x$ gets large. When $l(x)$ is linear it is called a slant or oblique asymptote.

No

Algebra

Chapter 1

Functions and their Applications

Section 7

More on Functions

Functions

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McMaster University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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02:10

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01:06

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02:50

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03:57

End behavior of rational f…

15:40

Suppose $f(x)=\frac{p(x)}{…

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If the rational function $…

01:36

we're told that we can do polynomial division um to get to write a rational polynomial. So uh polynomial divided by another polynomial. In terms of some if this has this has one degree higher than this, then we can write it as a linear function times at some other um rational polynomial with X in there than the denominator. And then this is going to have a lower order than this. And the fact is going to be two orders lower. Now. They ask us to take a look at this limit here and this lemon here as X goes to plus or minus infinity. So we can look at this function this here and that is simply the limit as X. Core the infinity of our over cuba. But we know that R is a lower order polynomial. Thank you. So this must go to zero. So that's what they asked us to show that this is zero. And likewise as we go to go to minus and feelings, we just have the limit of X goes to minus, infinity of our over Q. And again because our is a lower order polynomial Q, then this also close to zero. So we know that basically what it's showing us is that as X gets very large, this function looks like this function because as X gets very large, this thing gets small. So this looks like this. And that's only the case if if p is one degree higher than Q. Obviously if he has two degrees higher than we have in cubic function. Uh sorry quadratic function. Mhm. And we can we can do the same thing for any degrees here. As long as this degree of this one is greater than the degree of this one. We will have some kind of behavior as X goes to infinity. That will look like some other but look like a polynomial. And they will see lots of examples of those in the problems that in the common problems

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