Ok, in this video I just want to talk about finding the horizontal asymptotes of rational functions.
And finding horizontal asymptotes is very easy.
Again, not all rational functions have horizontal asymptotes.
One of two things has to happen. Either...
The degree of the numerator has to be exactly equal to the degree of the denominator,
Or the degree of the denominator is greater than the degree of the numerator.
In that... in those two cases, that's when we'll have horizontal asymptotes.
It turns out that if the degree of the numerator IS equal to the degree of the denominator...
The horizontal asymptote, the line "y=" were just gonna take the ratio of the leading coefficients.
If the degree of the denominator is GREATER than the degree of the numerator...
Then simply then simply the line, "y=0" is going to be a horizontal asymptote.
So, ok, let's look at the first example. It says which of the following have a horizontal asymptote? If so, let's state it!
Well hey notice the degree of the numerator is two, because of the X squared. The degree of the denominator is also X squared.
All we have to do is take the ratio of their coefficients, and it says the horizontal asymptote is going to be the line "y= 3/5" (three fifths)
Easy enough.
Uhh, notice in the next problem we have the degree of the numerator, and again the degree is just the highest power, the highest exponent.
We've got X to the 20th, we've got X to the 20th.
The degree of the numerator is equal to the degree of the denominator.
Again, we just take the ratio of the leading coefficients.
It says the line "y= 3/8" (three over eight) is gonna be our horizontal asymptote.
Notice in our last example, well now the degree of the numerator is 1, the degree of the denominator is 3.
Same idea, simply, uh, in this case we've got our second case, um, since the denominator has a degree larger than the numerator...
the horizontal asymptote is gonna be the line "y = 0"
Um
Certainly these are rules that you can just mechanically, uh, memorize.
I think you should really convince yourself why these things are, in fact, true.
Uhm
You really want to. You know, cause then it's not, then you're not memorizing it. Then you're really kind of understanding it.
So...
Uhm...
So...
So think about that.
Let's do two other examples, or a couple other examples here real quick.
So...
Suppose we have Y= X cubed minus 5 over X plus 4.
Suppose we have Y= uhm, let's say X plus 4 over X cubed minus 5.
Uhm...
Would these have horizantal asymptotes?
Again in this first example...
Well now, uhm, things are flip flopped.
Now the degree of the numerator is much, is, is larger than the degree of the denominator.
So... in this case.
Uhm...
Since the degree...
Of the numerator, is larger than the degree of the denominator.
In that case you have no horizontal asymptotes.
The graph is either gonna spike up to positive infinity or negative infinity as you go left AND right.
Ok?
And...
It doesn't have to do the same thing.
It could go to negative infinity on one side, and positive infinity on the other...
or vice versa.
Uhm...
Or both to infinity, or negative infinity. It really just depends on powers.
Ok.
So that goes with that one.
If we look at our other example here...
Again Y= X plus 4 over X cubed minus 5.
Well now the degree of the denominator IS larger than the degree of the numerator.
So the horizontal asymptote is gonna be the line "Y= 0"
Ok, so, again, all you're doing is if the degrees are the same, you're taking ratios of leading coefficients.
Again, leading coefficients are the numbers in front of the highest powered terms.
Uhm...
And...
That's it! Ok, so these are very useful little tricks.
Uhm...
Finding horizontal asymptotes is certainly something which you'll do fairly often in algebra, pre-calc, calculus.
So...
I think definitely useful tricks worth remembering.
So...
Alright! I hope this helps.
Uhm...
If you have any questions are comments, as always, feel free post em'.