Horizontal Asymptote For Rational Equations
Moderators: gmalivuk, Moderators General, Prelates

 Posts: 31
 Joined: Fri Mar 16, 2018 4:34 pm UTC
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^2x6) seems to easily disprove that. Am I doing something wrong, or is what the teacher said invalid?
Re: Horizontal Asymptote For Rational Equations
How are you figuring that (x+1)/(x^2x6) doesn't have an asymptote at 0?
gmalivuk wrote:Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one metame to experience both body's sensory inputs.

 Posts: 31
 Joined: Fri Mar 16, 2018 4:34 pm UTC
Re: Horizontal Asymptote For Rational Equations
Sizik wrote:How are you figuring that (x+1)/(x^2x6) 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
 Dr. The Juggernaut of Touching Himself
 Posts: 5387
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
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.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Horizontal Asymptote For Rational Equations
Mark_Cangila wrote:Sizik wrote:How are you figuring that (x+1)/(x^2x6) 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
 Factoids are just Datas that haven't grown up yet
 Posts: 5965
 Joined: Tue Nov 28, 2006 9:50 pm UTC
 Location: Sydney
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.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

 Posts: 31
 Joined: Fri Mar 16, 2018 4:34 pm UTC
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
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^2x6) 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)/(1000^210006) = 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.

 Posts: 31
 Joined: Fri Mar 16, 2018 4:34 pm UTC
Re: Horizontal Asymptote For Rational Equations
That explains some of the rules my teacher gave. Thanks.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5387
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Horizontal Asymptote For Rational Equations
yeah simple test cases are great for that kind of thing
totally aces
totally aces
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Who is online
Users browsing this forum: No registered users and 5 guests