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Voiceover:Right over here, I have the graph of f of x, and what I want to think about in this video is whether we could have sketched this graph just by looking at the definition of our function, which is defined as a rational expression. We have 2x plus 10 over 5x minus 15. There is a couple of ways to do this. First, you might just want to pick out any numbers that are really easy to calculate. For example, what happens when X is equal to 0? We could say f of 0 is going to be equal to, well, all the x term is going to be 0, so you're going to be left with 10 over negative 15, which is negative 10/15, which is negative 2/3. You can plot that one. x equals 0. f of x or y equals f of x is negative 2/3, and we see that point, let me do that in a darker color, you see that point right over there, so we could have plotted that point. We could also say, "When does this function equal 0?" Well, the function is equal to 0 when ... The only way to get the function equal to 0 is if you get this numerator equal to 0, so you could try to solve 2x plus 10 is equal to 0. That's going to happen when 2x is equal to negative 10. I just subtracted 10 from both sides. If I divide both sides by 2, that's going to happen when x is equal to negative 5. You see that, you see this right over here. When x is equal to negative 5, the function intersects the x-axis. That's just two points, but that still doesn't give us enough to really form this interesting shape over here. You could think about what other functions have this type of shape. Now what I want to think about is the behavior of the function at different points. First, I want to think about when this function is undefined and what type of behavior we might expect for that function when it's undefined. This function is going to be undefined. The only way I can think of to make this undefined is if I make the denominator equal to 0. We don't know what it means to divide by 0. That is undefined. The function is going to be undefined when 5x, let me do this in blue, when 5x minus 15 is equal to 0, or adding 15 to both sides, when 5x is equal to 15, or dividing both sides by 5, when x is equal to 3, f is undefined. Now, there is a couple of ways for a function to be undefined at a point. You could have something like this. Let me draw some axes right over here. Let's say that this is 3. You could have your function, it could look like this. It could be defined. It might approach something but just not be defined right at 3 and then just keep on going like that, or the other possibility is it might have a vertical asymptote there. If it has a vertical asymptote, it's going to look something like this. It might be approach, it might just pop up to infinity, and it might pop down from infinity on this side, or it might go from negative infinity right over here. That's what a vertical asymptote would look like, that as we approach from the left, the graph is approaching a vertical, but it never quite gets to x equals 3, I guess one way we could say it, or the function is not defined at x equals 3. As you approach from the right, the same thing, the function just, in this case, drops down. It almost becomes vertical. It's approaching negative infinity as x approaches 3 from the positive direction. So how would we know? Obviously, when you look at here, when we know the graph ahead of time, and if you say, "OK, this is easy." This is x equals 10. Let's see how many. This is 1, 2, 3, 4, 5, so each of these are 2, so x equals 3 is right over here. When you look at the graph, if you had the graph in front of you, you would see, "Oh, look, this is indeed a vertical asymptote." Just looking at the graph, you see that you have a vertical asymptote at x equals 3, and let me write that down, vertical asymptote, vertical asymptote, asymptote at x equals 3, at x equals 3, but how would you have known that? How would you have known that if you didn't have the graph here, if you just had this? We know it's not defined at 3, but how do we know it's not a point discontinuity and not instead a vertical asymptote? There's a couple of ways to do it. One way is you could try values near 3 and see what happens. For example, you could get your calculator out, and you could try, let's say, 3.01. If you say 2 times 3.01 plus 10, actually, oh, that's the numerator, and then I'm going to divide that, the numerator, by 5 times 3.01 minus 15. It gets us a fairly large number. It's exploding on us. If we got even closer, so if we did 2 times 3.001, plus 10 divided by 5 times 3.001, now I'm trying x is equal to 3.001 minus 15, we see we get even a larger number. As x gets closer and closer to 3, f of x seems to be exploding. It seems to be approaching positive infinity. That's one way to say, OK, this looks like, at least from this side, we are approaching, we are approaching positive infinity, so we would have been able to draw something like that, and then you could have tried values below. You could have tried values below, so you could have said ... Actually, let me just put the last entry here, and let me just change the 3.001s to 2.999, 2.999, and whoops, let me go over here. We have 2.999, and we get ... We're going really negative now. We're approaching negative infinity. If you just tried that out, that would give you a pretty good indication that the graph is looking something like that right over here, which also seems to match connecting these two points that we've already thought about. But now let's see what's going on as x approaches really large values or really positive values or really negative values. It looks like there is a horizontal asymptote here. Just looking at the graph, it seems like there is some value that as x approaches really large values, really positive values, f of x is going to be approaching that value, that asymptote from above. As x becomes really negative, it looks like f of x is approaching that from below. But how would we be able to figure that out just by looking at this? One thought experiment is just to say what happens to f of x as x approaches infinity? Let me write that down. As x approaches infinity, then f of x is going to approach what? As x approaches larger and larger values, the positive 10 and the negative 15 start to matter a lot less. At the highest degree term, the numerator and denominator start to dominate. We could say as x approaches infinity, f of x is getting closer and closer to 2x over 5x, which is 2/5, which is equal to 2/5. You could say f of x is approaching 2/5. If you really want to see that a little bit more concretely, let's imagine different values for x, as x gets larger and larger and larger. If we have, so x, f of x. If x is 1, then f of x is just going to be 2 plus 10 over 5 minus 15. Here, the 10 and the subtracting the 15 matter a lot. But if x were 1,000, then f of x would be 2,000 plus 10 over 5,000 minus 15. Now, the 2,000 and 5,000 are really setting the agenda. Then if x were, let's say, 1 million, 1 million, and I just use blue for more contrast, then f of x would be 2 million, 2 million plus 10, let me move over to the right a little bit, 2 million plus 10 over 5 million minus 15. Here, the 10 and the 15 are almost inconsequential. You can imagine if x were a billion or a trillion or a google, then the 10 and the negative 15 start to matter a lot and lot less. As x approaches infinity, these matter less. The highest degree terms matter, so f of x is going to approach 2x over 5x, which is 2/5. So f of x is approaching 2/5, and that's what this line looks like. 2/5 is same thing as 0.4, so f of x, and we see that in the graph. f of x is approaching that, but it's not quite getting close to it. It's not quite getting there. It's getting closer and closer to it as x goes to infinity, but it's not quite getting there because you're always going to have that plus 10 and that minus 15 there, so you're never going to be exactly at 2/5. The same thing is happening as x gets more and more and more negative. You can make all of these negative values. If this were negative 1, that would be negative 2, negative 5. If this were negative 1,000, it would be negative 2,000 over negative 5,000. If it's negative 1 million, it would be negative 2 million over negative 5 million. But you see, even in this case, f of x is approaching 2x over 5x, which is approaching 2/5, or you could say it's approaching negative 2 over negative 5 which is still 2/5, and you see that right over here. We would say that this function has a horizontal asymptote, horizontal asymptote, at y equals a horizontal line right over here, y is equal to 2/5. Hopefully, this graph here is helping us appreciate what these vertical and horizontal asymptotes actually are. But if we didn't have the graph, we could have said, OK, we're undefined at x equals 3. We could test some values around it, and so we could say, OK, look, it does look like we're approaching negative infinity as x approaches negative 3 from the left. It looks like we're approaching positive infinity as x approaches negative 3 from the right, so we could draw that blue point right there. We could graph these two points, when does f equal 0, What happens to f when x equals 0, and then we could think about the behavior as x approaches infinity or negative infinity, as x approaches infinity or negative infinity, and draw this horizontal asymptote. Between all of those, that would have been a pretty good way to be able to sketch this actual graph.