 So, where there's horizontal, there must also be vertical. Horizontal asymptotes occur if y goes to k as x goes to plus or minus infinity. But what if we switch things around? What if x goes to h when y goes to plus or minus infinity? In this case, we obtain a vertical asymptote. So, we might try to describe the end behavior as y goes to infinity in y equals 1 over x. Now, let's try and rearrange our equation to make the analysis a little bit easier. From y equals 1 over x, we'll multiply through by the denominator x, and that gives us yx equals 1. Now, we know y is going to infinity, so we'd like to be able to say something about x, so let's solve for x by dividing by y, and that tells us that x is equal to 1 over y. As y goes to plus or minus infinity, x is going to be 1 divided by a very large positive or negative number, which means that x is going to go to 0. And so, we might summarize as y goes to plus or minus infinity, x goes to 0. So, this tells us that x equals 0 is a vertical asymptote. Generally speaking, we're given an equation for y, and it's not easy to solve that equation for x. But to remember what we're looking at, if our graph has our points going to some specific value of x as y goes to plus or minus infinity, then we can read this as follows either. As y goes to plus or minus infinity, x goes to h, but also we could read this as x goes to h, y goes to plus or minus infinity. And this leads to the following important idea. If x going to a makes y go to plus or minus infinity, then x equals a is a vertical asymptote. Let's introduce some more adjectives. Another useful adjective is slightly more or slightly less. We'll use a superscript plus to indicate slightly more than a, and a superscript minus to indicate slightly less than a. And by this notation, we can now consider what happens as x goes to a, but always stay slightly more than a. We sometimes say that x is going to a from above. And likewise, we can consider what happens as x goes to a, but stays slightly less than a, which we call from below. And one very important thing to remember, the plus and minus in our notation has nothing to do with the sign of a. So let's try to describe the behavior of y equals one over x minus three as x goes to three, but stays slightly more than three, and as x goes to three, but stays slightly less than three. So remember the first thing we should do any time we have any rational function is to identify the forbidden values. And so here our denominator is x minus three. So we require x not equal to three. So suppose we let x go to three from above. Remember this means that x is a number that's around three, but it's a little bit more than three. So let's unpack our adjectives. If x is getting close to three, but staying slightly more than three, x minus three will get close to zero, but stay slightly more than zero. So we'll indicate that using a superscripted plus. Now zero plus is a small positive number. It's small because it's close to zero, and it's positive because it's slightly more than zero. So this means that one over x minus three will be a large positive number. So this means that as x gets close to three from above, one over x minus three goes to infinity, but equals means replaceable. So if we know what one over x minus three is doing, we also know what y is doing. And so we can say that as x gets close to three from above, y goes to infinity. On the other side, if we let x get close to three from below, in other words, we're going to let x get close to three, but stay slightly less than three, then x minus three gets close to zero, but it stays slightly less than zero. So that's a small negative number. And so one over x minus three will be a large negative number. So as x gets close to three from below, one over x minus three goes to minus infinity equals means replaceable. So y goes to minus infinity. So remember, a vertical asymptote is going to occur for any value of x that makes y go to plus or minus infinity. So that says that x equals three is a vertical asymptote. This suggests the following idea. Suppose we have a rational function where we've reduced it as much as possible. The asymptotes of the graph of y equals f of x will occur at x equals a, where q of a is equal to zero. In other words, they're going to occur at places that make the rational function undefined. So let's see if we can find the asymptotes of this rational function. And whether or not you're looking for asymptotes, the first thing we should always do is to find where the rational expression is undefined. And those will be places that make the denominator equal to zero. So we'll solve and we find that x equals one or four are solutions. So we require x cannot be equal to one, x cannot be equal to four. Next, we'll try to simplify. We can hope to cancel, but remember you can only cancel if both numerator and denominator are products. So this means we'll have to factor. Since we know the roots of x growth minus five x plus four, we know that it has to factor as x minus four times x minus one. And then we can factor our numerator and we can try to simplify. Since x equals one makes the denominator of the reduced expression equal to zero, then x equals one will correspond to a vertical asymptote. And we might summarize our work. In simplified form, our rational expression is y equals x plus one over x minus one, still four x not equal to one, x not equal to four. And since x equals one makes the denominator of our reduced expression equal to zero, there's a vertical asymptote at x equals one. One important idea to keep in mind while we still have to require x can't be one, x can't be four, only x equal to one gives us a vertical asymptote. The vertical asymptotes only depend on the simplified form. Now you might wonder what happened to that x equal to four. In order to answer that question, you'll have to take calculus.