Horizontal Asymptotes - Further Detail

Horizontal Asymptotes Further Detail
Hi, everyone!
Welcome back to integralcalc.com.
We’re going to be talking a little bit more about horizontal asymptotes today.
I just did another video that was kind of the basics of horizontal asymptotes, finding them and graphing them.
But I wanted to go a little more in-depth because I think it's really helpful to understand why these horizontal asymptote rules are what they are.
I think it helps you to understand them better and then you'll be quicker at them when you have to figure it out on a test.
so you can kind of double check them by looking at the graph and saying ‘Okay, yes that makes sense,’
based on my understanding of how you actually find horizontal asymptotes.
So I wanted to look at these rules and show you why we can prove that they are what they are.
We said in the last video that these were our rules and I have rational functions here, three different rational functions,
simplified to just the first term of the rational function in both the numerator and the denominator.
And in the last video, we talked about how we're only looking at the exponent on this first term of the numerator and denominator of the rational function.
So how T stands for top and B stands for bottom.
So in this case, the top is equal to the bottom as in this example, two equals two,
when the top is less than the bottom, three less than four;
and when the top is greater than the bottom, three greater than two.
And how when we have these three different kinds of relationships,
we know that the horizontal asymptotes are respectively.
In this case, the coefficients on the first two terms, so here we have three over four;
which is definitely the horizontal asymptote in this case, when the top is equal to the bottom here.
In this case when the top is less than the bottom, the horizontal asymptote is the x axis, also known as the line, y equals zero.
And then in this case, when the exponent on the top is greater than the exponent on the bottom, we have no horizontal asymptotes.
But what I want to show you is why that's true.
So let's take the first case here.
And I've set all of these equal to y, right? Because we'll always have f of x equals a rational function, some function.
Same thing as, you know, y equals the function.
So in this case... Well, in all cases with horizontal asymptotes, no matter where your horizontal asymptote is on the graph,
basically what's happening is the graph is leveling out towards a value as you go farther and farther out to the right or to the left on the graph.
So you know, it’s getting closer and closer and closer to that same value.
That value is the limit of the function as f of x goes...
as x goes to positive infinity or to negative infinity because the graph is getting closer and closer and closer to that line.
So essentially, when x is ten million or when x is negative ten million,
the graph is almost at that asymptote so we can basically say the value is whatever that asymptote is.
We have this graph leveling out and what we want to do is find a relationship between x and y so that we can see what the asymptote is.
In this case, what we're going to end up doing is just simplifying these basic functions.
If we simplify this function, we have x squared over x squared.
So those two will cancel, right? And we'll just be left with y equals three-fourths.
So, BOOM! We're one-third done. That's really obvious that our horizontal asymptote is going to equal y equals three over four.
When we simplify the function, that's all we have left.
So when the exponent on the top is equal to the exponent on the bottom, we can just simplify and drop those terms and we're left with these three-fourths and that's our horizontal asymptote.
If we go now and simplify this function here, we have y equals x to the third over x to the fourth, we would end up with y equals one over x, right?
The x to the third would cancel on the top and we'll be left with one x on the bottom here.
So y equals one over x. So now we need to figure out what the asymptote is.
You can imagine, as x gets very, very, very, very, very large, or very, very, very, very negative, this number here, one over x is going to get very small.
So if x is one... then...
And we're basically just plotting points, right?
So if x is one, y will also be one because we'll have one over one.
If x is ten, then y would be one-tenth.
If x is one hundred, then y will be one over one hundred.
So you can see, the y value as x gets larger is getting smaller and smaller and smaller;
which, if we are graphing,as x gets larger, one, ten, one hundred... As x gets larger this way, y gets smaller.
So the graph of this might look like... Here's the x axis so... It might look like that, right?
The graph is going to approach the x axis in both directions whether you get really positive or really negative.
You can see as well if x is negative, we're going to have the same trend.
So if you simplify this function and then you plot some points,
you'll see that y will get very very small to the point where it's point zero zero zero zero zero one,
which is as close to the x axis as you can get for y value, without y actually equaling zero.
y equals zero is the x axis here.
So the farther you go out in either direction, the smaller y is going to get and the closer it’s going to get to the x axis.
Therefore if you go out as far as possible all the way to infinity, basically the limit of the function y equals zero.
That’s your horizontal asymptote or the x axis.
No matter what you might have in front of these in terms of coefficients, or what the exponents are here,
as long as the top exponent is less than the bottom exponent, the x axis will end up being your horizontal asymptote.
Those are the first two and then the last case where the exponent on the top is greater than the exponent on the bottom.
If we simplify this function, we'll end up with y equals x over one
because this x squared will cancel and we'll be left with x on the top, which of course is just y equals x.
We can get rid of that. y equals x.
So if we plot that graph, as I'm sure you can see...
So if we plot that graph, if x is one, y is one.
If x is ten, y is ten.
If x is one hundred, y is one hundred, right? Because we've just got y equals x.
Similarly, if x is negative one, y is negative one.
If x is negative ten, y is negative ten.
And if x is negative one hundred, y is negative one hundred.
So what we end up with if we plot a line here is basically, this line. y equals x.
As you can see, that will go on forever in this direction and forever in this direction.
And we have no horizontal asymptote because no matter where you try to intersect the graph,
it's never going to level off either way so it will never come down here and start approaching this line.
This one will never level-out here and start approaching a line.
It's just going to go in a straight line up to the right and down to the left because we have a y equals x relationship.
So there is no horizontal asymptote and that's why we know that this is true.
So if you are doing a problem on a test, it's obviously great to memorize these rules
because then you're just quick with it and you can spit it out on the test without having to waste any time.
But if you are unsure and you want to double check yourself,
the easiest way is just to take the first term in the numerator and the first term in the denominator, get rid of everything else and simplify.
Here, you can cancel. Here, you can see the graph approach the x axis.
and here, you can see that we end up with this line, leaving you with no horizontal asymptotes.
So I hope that helps kind of explain and give you a picture on your head about why these rules are true.
But obviously, my suggestion would be memorize and if you're unsure, graph this out and use that to help you.
Thanks for watching, I hope that helped. And I'll see you guys next time.
Bye!