Horizontal Asymptote For Rational Equations

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Mark_Cangila
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Horizontal Asymptote For Rational Equations

In my Algebra 2 class the teacher said that for rational expressions where the numerator has a lesser degree than the denominator, the horizontal asymptote is 0. However, I seem to have found an exception to that. The equation (x+1)/(x^2-x-6) seems to easily disprove that. Am I doing something wrong, or is what the teacher said invalid?

Sizik
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Re: Horizontal Asymptote For Rational Equations

How are you figuring that (x+1)/(x^2-x-6) doesn't have an asymptote at 0?
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Mark_Cangila
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Re: Horizontal Asymptote For Rational Equations

Sizik wrote:How are you figuring that (x+1)/(x^2-x-6) doesn't have an asymptote at 0?

This is what I did:
First, 0 over any number other than 0 is 0. The numerator is 0 when x = -1. If you plug in -1, it evaluates to 0/-4. That is 0. Therefore, at -1 the equation evaluates to 0, so it can not have an asymptote at 0.

doogly
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Re: Horizontal Asymptote For Rational Equations

It can cross the x axis, do some stuff, and then come back around to it asymptotically. You're correct about where it crosses, but the conclusion that it can't also be an asymptote is wrong.
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cyanyoshi
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Re: Horizontal Asymptote For Rational Equations

Mark_Cangila wrote:
Sizik wrote:How are you figuring that (x+1)/(x^2-x-6) doesn't have an asymptote at 0?

This is what I did:
First, 0 over any number other than 0 is 0. The numerator is 0 when x = -1. If you plug in -1, it evaluates to 0/-4. That is 0. Therefore, at -1 the equation evaluates to 0, so it can not have an asymptote at 0.

A horizontal asymptote is just a horizontal line that a function stays arbitrarily close to for all large enough "x". It's fine for the function to intersect its asymptote along the way.

jestingrabbit
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Re: Horizontal Asymptote For Rational Equations

For instance, its true to say that sin(x)/x has a horizontal asymptote of 0. You just can't stay on the line if its an asymptote.
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Mark_Cangila
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Re: Horizontal Asymptote For Rational Equations

jestingrabbit wrote:For instance, its true to say that sin(x)/x has a horizontal asymptote of 0. You just can't stay on the line if its an asymptote.

Thanks! I think I understand it now

LaserGuy
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Re: Horizontal Asymptote For Rational Equations

Mark_Cangila wrote:In my Algebra 2 class the teacher said that for rational expressions where the numerator has a lesser degree than the denominator, the horizontal asymptote is 0. However, I seem to have found an exception to that. The equation (x+1)/(x^2-x-6) seems to easily disprove that. Am I doing something wrong, or is what the teacher said invalid?

An test sanity check to test this, is try putting in a very large value for x. It doesn't even need to be that large. Like 1000 or so will work for your equation:

(1000+1)/(1000^2-1000-6) = 1001/998994 ~ 0.001.

You can see pretty clearly here that it's heading toward zero in this direction. Most of the interesting behaviour in functions happens near their zero crossings; the dominant behavior in such equations far away from the crossings is always going to depend on the leading term in the expression. So in your case, in the numerator, x >>> 1 and in the denominator, x^2 >>> x >>> 6. So as x approaches infinity, the behaviour of this equation will resemble x/x^2 = 1/x (where I first drop all of the secondary terms, then simplify), which, for large x, is zero.

Mark_Cangila
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Re: Horizontal Asymptote For Rational Equations

That explains some of the rules my teacher gave. Thanks.

doogly
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Re: Horizontal Asymptote For Rational Equations

yeah simple test cases are great for that kind of thing
totally aces
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