So we're going to look at two very specific graphs for a moment. These are graphs that you've actually been putting up on the boards during workshop when you're doing some prerequisite review. We're looking at what I call the reciprocal functions. So this would be a graph of the reciprocal function where the power in the denominator is an odd number.
All right. And if you remember, this is because when you. Raise positive x values to an odd power, they stay positive. And when you raise negative numbers to an odd power, they stay negative. And obviously, if x is equal to 0, we have discontinuity.
We have a problem in our domain, so we're going to have a vertical asymptote at x equals 0. And there's no way to divide 1 by any number and get 0. So the y values for all of these. will never be zero. So we'll have a horizontal asymptote at y equals zero. So this is our one over x.
One over x cubed will look similar. It'll be a little steeper. One over x to the fifth, one over x to the fourth. And then one other thing that I just wanted to make sure to note here is that n also has to be a positive number in order for this to work.
Because if n is actually a negative number, raising x to a negative number in the denominator is actually x to a positive number in the numerator. And then we're talking polynomial functions where we have no asymptotes. All right. So let's make sure that we remember this graph. While I'm looking at it, I want you to think about the fact that as my x values are getting very, very large, my graph is decreasing and it's getting very, very close to the x-axis.
And the same is happening over here. as my x values are going towards negative infinity, my graph is getting closer and closer to the x-axis and approaching zero. So actually what we're looking for here is as x approaches either positive or negative infinity, my function is approaching y equals zero.
Remember, it never has to quite get there. It just has to be getting arbitrarily close. All right. So we are going to want to see these functions in class during the next lecture.
Okay, this is the type of function we want to see because we know what the graph of the function is doing as we approach infinity or negative infinity. All right, similar function, but now when n is even, all right, so now if n is even, again, raising positive numbers to an even power keeps them positive. But raising negative numbers to an even power is going to make them positive. And so the graph goes. both in the positive y in either hemisphere.
Excuse me. All right, same asymptotes for this graph. Okay, again, this is only true if n is positive.
Okay, if n was negative, again, we'd have 1 over x to a negative power, which means that negative power would actually come up into the numerator as a positive power. All right, and again, As x approaches positive infinity, as these x values are getting larger and larger, this graph is tending toward the x-axis. So it's actually approaching zero for the y values and the same thing in the opposite direction.
Okay. So we really do want to see this type of function in our next lecture, which is going to be limits at infinity. All right.
So this is where we want the 1 over x to the n form. And what you're going to be seeing in class is we actually are starting with rational forms. OK, so these rational functions that I've written here are great, but I really would like to see this form somewhere in there.
OK, so we have to figure out a way to bring that in. So the way you're going to do this is you're going to look for the highest power of X in the denominator, always in the denominator. OK. So we're going to locate the highest power in the denominator. See the background nice and straight.
And I'm trying to find another color, actually. OK, so for this situation, I see x squared is my highest power of x in the denominator, my highest power function. OK, so what I'm actually going to do is sort of a similar thing to other algebraic manipulations we've been working with. I am going to multiply my numerator and denominator by the reciprocal function with that highest power. So 1 over x squared.
And then I'm also going to multiply by 1 over x squared in the denominator. Okay. So what that means is everything in my numerator will get multiplied by 1 over x squared.
And everything in my denominator will get multiplied by 1 over x squared. Now, keep in mind, that's the same thing as saying everything in my numerator will be divided by x squared. And everything in my denominator will be divided by x squared. So let's work through that really quick.
And I'll show an intermediate step in there that you may not need to show. You might be able to get through this without showing this next step, where I say x squared times 1 over x squared is the same thing as x squared over x squared. And then 5x times 1 over x squared is the same thing as 5x over x squared.
And then 1 times anything is just that thing, so 1 over x squared. All right, and then that whole denominator is now going to be x over x squared and 4x squared over x squared. And then what we need to do is simplify.
All right, so each of these rational functions, again, we're trying to get this form to show up in all of our terms, or at least most of our terms, okay? If I take x squared and divide it by x squared, that is equal to 1 for x not equal to 0, but I'll be able to still use that same domain. All right, because when you see the final result, you'll see that x can't be 0 in a number of these terms.
x over x squared reduces, and I can rate this term as 5 over x, because when I divide two things in the same base, I subtract their exponents. So here I'm doing 2 minus 1, all right, but because the higher power is in the denominator, that's where it's going to stay. I can't do anything about this term here, right?
There's nothing better to do with that. x over x squared simplifies the same way as this term did here. So 1 over x, I just don't have a coefficient of 5 on it.
And then 4x squared divided by x squared is just 4. And now you see you've got that reciprocal function form that you want in one, two, three terms. And the other two terms are constants. And you'll remember that earlier we said x can't be zero. Well, we're good to go, right? This still says the same thing.
So this is actually going to be your helpful form for finding limits at infinity. All right. So let's go ahead and take a look at this second example over here.
And I'm looking for the highest power of x in my denominator. And I see x to the fourth. Okay, so I am going to multiply my numerator and denominator by 1 over x to the fourth divided by 1 over x to the fourth.
All right. And again, we're going to go ahead and distribute into the numerator. All terms get multiplied by 1 over x to the fourth.
That is the same thing as saying all terms divided by x to the fourth. So we're distributing into the numerator and the denominator. All right, again, this intermediate step here you may not need to show. You might be able to simplify in your head, and that's totally fine. We'll have 3x squared divided by x to the fourth minus 4x cubed divided by x to the fourth.
And then all of that divided by x to the fourth divided by x to the fourth. And then minus 2x squared divided by x to the fourth. All right. And the same simplification we're looking to when we divide two things of the same base, we're going to subtract their exponents. 4 minus 2 leaves me with a power of 2 in this denominator.
So 3 over x squared minus. 4, 4 minus 3 is 1, so 4 over x to the first divided by x to the fourth divided by x to the fourth is 1. And then again, 4 minus 2 is 2, so we're going to have 2 over x squared. And again, I was able to achieve 1, 2, 3 terms that fit my reciprocal 1 over x to the n form, and a constant term, which is always great too.
So this is going to be the technique, the algebraic manipulation that you're going to need for limits at infinity.