 An important idea from calculus is end behavior and asymptotes. Now, this is actually a calculus idea, but it's frequently covered in pre-calculus, which is why it's here. And that comes back to the following idea. An important property of a graph is its end behavior. What happens to the graph as you go very far to the right, or very far to the left? Well, right and left are geometric ideas, so we can express these geometric ideas using the algebraic symbol infinity. And we can talk about it as follows. The graph consists of the points x, y. As we go to the far right, our x-coordinate goes to infinity. And as we go to the far left, our x-coordinate goes to minus infinity. So this geometric question about end behavior becomes the algebraic question, what happens to the values of x, y as x goes to infinity? And it'll be helpful to be able to compute with adjectives. And what that means is the following. We're concerned primarily with two features of a number, the sign and the magnitude. The sign of a number is either positive or negative. The magnitude of a number is either large or small. And an important idea here is to either go big or go small. And so we might classify the numbers 3148734 or negative 0.0000001. So let's remind ourselves of what those adjectives are. And so 3148734 is large and positive. Our second number is small and negative. The reason that it's useful to do this is that when dealing with end behaviors, we can focus on the sign and magnitude. And it will be helpful to be able to perform computations on sign and magnitude. What do we mean by that? Well, let's try it out. Let's describe what happens to principal square root of x as x goes to infinity. So let's take that apart. Since x is going to infinity, we can think about x as being a large positive number. So now if x is a large positive number, then the principal square root of x will be positive because we're looking at the principal square root of x. And if x is large, the principal square root of x will also be large. So principal square root of x goes to a large positive number, which we can express as going to infinity. What about the end behavior of x cubed? Since x is going to infinity, once again, we can think of x as a large positive number. And so x cubed will be large and positive. And so we can say x cubed goes to infinity as well. Since I'm planning on running for politics, I'd like you to believe me on less and less evidence. So on two pieces of evidence, namely what happens to square root of x as x goes to infinity, and what happens to x cubed as x goes to infinity, we might claim the following. Suppose n is greater than zero. Then as x goes to infinity, x to power n also goes to infinity. And since I'm an aspiring politician, I want you to be convinced by as little evidence as possible. Believe the men behind the curtain. But if you want to be a good mathematician or a good citizen of a democracy, one of the things you should always do is seek additional evidence. This claims something is true if our exponent is greater than zero, we might see what happens as x goes to infinity to x squared, x to the fifth, cubed root of x, and so on. What about in the other direction? What happens if x goes to minus infinity? So again, this corresponds to the geometric idea of going to the way, way, way, way, way, way, way, way, way left. And so if x goes to negative infinity, then we can think about x as a large negative number. Now since x cubed is x times x times x, we're multiplying an odd number of negative numbers together. So the end result is a negative number. At the same time, all of the factors are large, so the product will be large. So x cubed will be a large negative number. And so that means that as x goes to minus infinity, x cubed will also go to minus infinity. What about x squared? So if x goes to minus infinity, then x is a large negative number. And since x squared is x times x, we're multiplying an even number of negative numbers together. So the end result is a positive number. Since all of the factors are large, the product will be large. So x squared will be a large positive number. So this means that as x goes to minus infinity, x squared will go to positive infinity. What about something like 1 over x cubed? So as x goes to infinity, x cubed also goes to infinity. And now let's consider the expression 1 over x cubed has us divide 1 by a large positive number. When we divide 1 by a positive number, we get a positive number. When we divide 1 by a large number, we get a small number. So 1 over x cubed will be a small positive number. And we might say that 1 over x cubed goes to zero. In the other direction, as x goes to minus infinity, x cubed also goes to minus infinity. The expression 1 over x cubed has us divide 1 by a large negative number. When we divide 1 by a negative number, we get a negative number. But when we divide 1 by a large number, we get a small number. And so 1 over x cubed will be a small negative number. And we say that 1 over x cubed goes to zero, where the fact that we end up with a negative small number doesn't really matter. A negative small number is still very close to zero. And so, as part of our political run, we'll try to convince you on the basis of one piece of evidence. Let n be greater than zero, then as x goes to infinity, 1 over x to the n goes to zero, and likewise if x goes to minus infinity. Again, don't be taken in, don't be convinced by one piece of evidence. We might try to find out what happens if I have 1 over x squared, 1 over x to the fifth, 1 over square root of x, and so on. Always look for the evidence.