 This notion of end behavior leads to the idea of a horizontal asymptote. This emerges as follows. If we graph an equation, the end behavior of the expressions tell us something about what happens very far away from the origin. Suppose that as x goes to infinity, y goes to k. Then, as we go to the far, far, far, far, far right, the points x, y will get close to the line y equals k. And we say that y equals k is a horizontal asymptote. Similarly, if as x goes to minus infinity, y goes to h, then y equals h is a horizontal asymptote. So, for example, find the end behavior of 1 over x cubed and interpret in terms of asymptotes of the graph of y equals 1 over x cubed. So, as x goes to infinity, 1 over x cubed goes to 0. And equals means replaceable. So, if y equals 1 over x cubed, y also goes to 0. And that means the line y equals 0 is a horizontal asymptote. And so, if we follow our graph of y equals 1 over x cubed to the far, far, far, far right, our points will approach y equals 0. What happens in the other direction? As x goes to minus infinity, 1 over x cubed goes to 0. So, again, y equals 0 is a horizontal asymptote. And so, once again, if we follow our graph to the far, far, far, far left, our y coordinates will approach 0, and our graph will approach the line y equals 0. Now, but a more general rational expression. So, an idea to keep in mind here is that it's easier to solve a lot of little problems than one big problem. So, what can we do? If we divide numerator and denominator by the highest power of x in the denominator, we'll create a lot of little problems. So, we see that in the denominator, our highest power of x is x squared. So, we'll divide numerator and denominator by x squared. Equivalently, we'll multiply by 1 over x squared. So, if we do that, we get... And let's simplify each of our terms. Now, we're interested in knowing what happens as x goes to plus or minus infinity. So, that means we should ask ourselves what happens to 7 over x, 8 over x squared, 4 over x, and 5 over x squared. As x goes to plus or minus infinity, these expressions go to... And so, the numerator of our expression is going to go to 1 plus 0 plus 0. And the denominator is going to go to 3 plus 0 plus 0. So, our expression is going to go to 1 third. So, y equals 1 third is a horizontal asymptote in both directions. How about something like this? So, we want to consider the end behavior of our function 3 plus 3x plus 7 over x squared plus 5x plus 7. So, we want to know what happens as x goes to plus or minus infinity. So, we want to divide numerator and denominator by the highest power of x in the denominator. And that power is x squared. So, if we multiply numerator and denominator by 1 over x squared, we get... Which we can simplify. And as x goes to infinity, 3 over x and 7 over x squared go to... 0 and 5 over x and 7 over x squared go to... 0 as well. And so, our expression goes to... 3. So, y equals 3 is a horizontal asymptote. Or, take something like this. So, again, we want to divide by the highest power of x in the denominator. So, that highest power is... x squared. So, multiplying by 1 over x squared, simplifying, and as x goes to plus or minus infinity, 3 over x and 5 over x squared and 18 over x squared all go to... 0, 100 stays 100, but this numerator x is going to go to plus or minus infinity. And what that means is that this whole expression is going to go to plus or minus infinity. And we could say that y is going to go to plus or minus infinity. More importantly, we make the observation that there are no horizontal asymptotes.