Calculus: Functions with Asymptotes and Holes

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Curve Sketching
Asymptotes
Functions with Asymptotes and Holes Page [1 of 1]
All right, next step let's take a look at this example: f(x) = . So I want to graph this, which, of course, is another question, and you know all of the many parts. Asymptotes, critical points, increasing and decreasing concavity points and inflection....
So the first step since I see is the rational function - since I see it as the denominator - what I will actually do is first find the asymptotes. So let me begin with the vertical asymptotes. How do you find the vertical asymptotes? What you do is you try to factor as best you can, cancel, and see when the bottom is equal to zero. So if I factor this, what do I see? I see (x + 3) (x - 2). And on the bottom, the difference of two perfect squares, (x + 3) (x - 3). I notice there is a cancellation I can perform. I could cancel these things away. But of course remember, if you cancel, you have to make a promise with me - you have to promise me that that term will never be zero. So, in fact, I have to say right now that we'll do that I promise you that never, ever, ever will x equal in this case negative three because negative three is what makes that zero. So as long as you make that promise, then we could cancel away, and then what am I left with? Well, as long as you abide by that rule, I can now write f(x) as .
I could now find the vertical asymptote, which would then be what? Well, where the bottom equals zero, which would be at x equals three, or I could remember that just moments ago this was the example we worked out. We just graphed this. In fact, let me remind you of what happened; and if you just jumped in the middle here, you might want to go back just one example back because here was our work for working through this exact same function. All of this great work, and we saw this picture. Well now what we discovered is that the function at hand is exactly equal to that function with the promise that x doesn't equal negative three. That's the only difference between this function and that function. So this function equals that function as long as x doesn't equal negative three. If x equals negative three, this is undefined.
So basically I can now just jump back to here and say, "What's the only difference?" Well, at negative three, I'm going to have a problem. Do I have an asymptote? No, I don't have an asymptote, I just have that point removed. So, in fact, I have a hole there. So really it looks like that, and I have a really fancy picture of this. Here's the fancy picture of the curve with the hole, and you can see it's the exact same picture as the previous example but at negative three we have that hole - that missing point.
So be careful. Just because the bottom equals zero doesn't mean you have an asymptote. You may just have a hole. That's why you have to cancel. Keep track of what you're canceling, and then see whether the bottom equals zero to find the asymptote here.
That was pretty easy. We'll try one more together, and then we'll move on. See you there.