Alpha Epsilon Delta

Transcription

In the last video, we tried to come up with a somewhat rigorous definition of what a limit is, where we say when you say that the limit of f of x as x approaches C is equal L, you're really saying-- and this is the somewhat rigorous definition-- that you can get f of x as close as you want to L by making x sufficiently close to C. So let's see if we can put a little bit of meat on it. So instead of saying as close as you want, let's call that some positive number epsilon. So I'm just going to use the Greek letter epsilon right over there. So it really turns into a game. So this is the game. You tell me how close you want f of x to be to L. And you do this by giving me a positive number that we call epsilon, which is really how close you want f of x to be to L. So you give a positive number epsilon. And epsilon is how close do you want to be? How close? So for example, if epsilon is 0.01, that says that you want f of x to be within 0.01 of epsilon. And so what I then do is I say well, OK. You've given me that epsilon. I'm going to find you another positive number which we'll call delta-- the lowercase delta, the Greek letter delta-- such that where if x is within delta of C, then f of x will be within epsilon of our limit. So let's see if these are really saying the same thing. In this yellow definition right over here, we said you can get f of x as close as you want to L by making x sufficiently close to C. This second definition, which I kind of made as a little bit more of a game, is doing the same thing. Someone is saying how close they want f of x to be to L and the burden is then to find a delta where as long as x is within delta of C, then f of x will be within epsilon of the limit. So that is doing it. It's saying look, if we are constraining x in such a way that if x is in that range to C, then f of x will be as close as you want. So let's make this a little bit clearer by diagramming right over here. You show up and you say well, I want f of x to be within epsilon of our limit. This point right over here is our limit plus epsilon. And this right over here might be our limit minus epsilon. And you say, OK, sure. I think I can get your f of x within this range of our limit. And I can do that by defining a range around C. And I could visually look at this boundary. But I could even go narrower than that boundary. I could go right over here. Says OK, I meet your challenge. I will find another number delta. So this right over here is C plus delta. This right over here is C minus-- let me write this down-- is C minus delta. So I'll find you some delta so that if you take any x in the range C minus delta to C plus delta-- and maybe the function's not even defined at C, so we think of ones that maybe aren't C, but are getting very close. If you find any x in that range, f of those x's are going to be as close as you want to your limit. They're going to be within the range L plus epsilon or L minus epsilon. So what's another way of saying this? Another way of saying this is you give me an epsilon, then I will find you a delta. So let me write this in a little bit more math notation. So I'll write the same exact statements with a little bit more math here. But it's the exact same thing. Let me write it this way. Given an epsilon greater than 0-- so that's kind of the first part of the game-- we can find a delta greater than 0, such that if x is within delta of C. So what's another way of saying that x is within delta of C? Well, one way you could say, well, what's the distance between x and C is going to be less than delta. This statement is true for any x that's within delta of C. The difference between the two is going to be less than delta. So that if you pick an x that is in this range between C minus delta and C plus delta, and these are the x's that satisfy that right over here, then-- and I'll do this in a new color-- then the distance between your f of x and your limit-- and this is just the distance between the f of x and the limit, it's going to be less than epsilon. So all this is saying is, if the limit truly does exist, it truly is L, is if you give me any positive number epsilon, it could be super, super small one, we can find a delta. So we can define a range around C so that if we take any x value that is within delta of C, that's all this statement is saying that the distance between x and C is less than delta. So it's within delta of C. So that's these points right over here. That f of those x's, the function evaluated at those x's is going to be within the range that you are specifying. It's going to be within epsilon of our limit. The f of x, the difference between f of x, and your limit will be less than epsilon. Your f of x is going to sit some place over there. So that's all the epsilon-delta definition is telling us. In the next video, we will prove that a limit exists by using this definition of limits.