Calculus: Vertical Asymptotes

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Curve Sketching
Asymptotes
Verticle Asymptotes Page [1 of 1]
When you have functions that have x's in the denominator, you may actually have asymptotes. Asymptotes sound pretty scary, almost obscene, but really if you take it slowly, they're not that big of a deal. So, I want to talk to you a little bit about asymptotes for a bit.
Okay. So what's an asymptote? Well, asymptotes are actually lines that the function heads toward, but never actually touches. Those are lines that the function inches up and butts up against very gently, but never touches with the additional rule that the function is getting larger and larger and larger or smaller and smaller and smaller.
The idea is that the function is moving in a particular direction, but heading off to a horizon, either going up, going down, going right, or going left as you approach the line. Let me show you some examples of vertical asymptotes.
Probably the most famous example is the tangent function that we looked at earlier, because these have vertical -- this function has vertical asymptotes at two pi and at three pi over two. And you'll notice that, in fact, this vertical asymptote is a vertical line, which has the feature that the function wants to nestle up against it and yet never touches it. This gets as close to this line as you can imagine.
In fact, if you give me any little offset, I will make sure, if I go out far enough, that this curve actually goes past that offset, gets even closer to the line. But notice the curve gets larger and larger and larger. It keeps growing. On this part, notice that it actually comes down and seems smaller and smaller and smaller, and, again, butts up against it.
Okay, now how do you find vertical asymptotes? Well, the trick to vertical asymptotes in the rational function example--let me show you another example here. Here's another example of a function--I'll let you guess what the function is--where there's a vertical asymptote right here and additional vertical asymptote here and here. And you can see that here the function is heading up toward that thing, and it's coming down toward that thing. The vertical asymptote is located right there. Similarly, there's a vertical asymptote here.
So, how do you find these things in practice? Well, the first thing you do when you have a rational function is see where the denominator equals zero. So, when you want to look at the rational function, the very first thing to do when the denominator's equal to zero is to make sure that you can't factor and cancel anywhere in the fraction.
So, to find where you have vertical asymptotes, you take your rational function, factor the top and factor the bottom. Then in cancellation, you make the cancellation, and then take a look at what's left. What's going to happen? Well, you look at where the bottom equals zero, because that's the where function will be undefined, okay? And then, you look and see what the top is. And the top will not be zero there. And so, therefore, you have a vertical asymptote.
So, vertical asymptotes are really easy to find. You factor the top. You factor the bottom. You cancel away anything you see in common, and then look at where the bottom equals zero. Take those values, and those are your vertical asymptotes. Let's look at some examples really fast.
Suppose we look at the simplest function that has a denominator, this one. To find a vertical asymptote, the first thing I do is see if I can cancel or factor. Well, no. This is so simple. There's no canceling or factoring I can do. This is one over x. That's the lowest term.
Okay, well, the next thing I ask myself is, "Okay, where is the bottom equal to zero?" Well, the bottom equals zero at x equals zero. So, the bottom equals zero at x equals zero. That means this is a vertical asymptote. See how easy that was? That means that in this graph, x equals zero, which is the vertical line represented by the y-axis, is going to be a line that this function is going to butt up against somehow. And you may remember what this function looks like. I'll draw it in, even though we haven't seen this yet, officially, but just to get a sense of this.
Shoot, I did that wrong. You would think I would know how to graph these functions. I got so excited. I'm going to do this again, because, in fact, we want this to go down. Sorry about that. It's good to see - isn't it good to see people make mistakes? First of all, it makes you feel better. And also, it illustrates the fact that it's not a big deal. Making mistakes is not a big deal.
Well, really what we want here is the function to go like this and down like that, and comes up, and you can see this is really a vertical asymptote. That x equals zero line is a vertical asymptote. Okay, let's take a look at another example.
This example--y equals x minus two divided by x minus three. Can I factor, cancel? No, there's nothing I can factor or cancel here, so, in fact, that's the end of the story. I look at where the bottom equals zero, and I see the bottom is zero exactly when x equals three. So, I have a vertical asymptote there--not a big deal. So, if you look at the graph of this at the height--at x equals three--the vertical line three units over from the y-axis--one, two, three--the function somehow is going to butt up against it.
And the last example I wanted to look at is y equals x squared plus x minus six divided by x squared minus nine. What would you do here? Well, the first thing--like I said--you have to try to do is factor. Now, in fact, this can be factored quite a bit. So, I factor this. What do I see?
Well, on the top I see the signs are going to be opposite, and they have to multiply to give six and combine to give plus one. So, I think I want a three here and a two here. And on the bottom, that's the difference of two perfect squares. So, I actually know how to factor that pretty well. It's x plus three times x minus three.
And if you look at that, you notice there is a cancellation we can make. We can actually cancel this piece with that piece. Now, we've got to go back to the early points of this course, and remember that the only time you're allowed to cancel is if you promise me you're not canceling away zero. So, I make this under the agreement that, in fact, this--x plus three--does not equal zero, which means I cancel, but now, you've got to promise me that x doesn't equal the number negative three. Because if x plus three is zero, that means that x equals negative three, and that's not allowed. So, you have to remember this.
But armed with that, we now cancel and we're left with x minus two divided by x minus three. Well, that's actually the same problem as the previous one. And we already did the work, and we see that there's a vertical asymptote at x equals three.
The important thing in this example is that one maybe tempted to think there's a vertical asymptote at x equals negative three because the bottom is zero there. But that's not the case. The vertical asymptote is at three, because these things cancel away, and this just provides a hole in the function. And if you're sort of shaky on that, I invite you to go back and revisit the discussion we had on parabolas, because in that discussion, we actually looked at graphs of functions that actually had little holes in them, and we cancelled things away under a proviso of this sort. You might want to take a look at that.
The important thing is to always factor if you can, cancel away the common factors on top and bottom, making sure the function never equals that. Those are holes. And then, set the bottom equal to zero, solve, and find all your vertical asymptotes.
Okay, up next, we'll take a look at horizontal asymptotes--what happens as you go off to the horizon--in this case, this way and that way. We'll see what happens. Up next, I'll meet you there.