AP®︎ Calculus AB (2017 edition)
Some background intuition to make the formal definition of a limit make intuitive sense. Created by Sal Khan.
Let's try to come up with a mathematically rigorous definition for what this statement means. The statement that the limit of f of x as x approaches c is equal to L. So let's say that this means that you can get f of x as close to L as you want. I'll put that in quotes right over here, because it's kind of a little loosey goosey as how close is that. But as close as you want by getting x sufficiently close to c. So another way of saying this is, if you tell me, hey, I want to get my f of x to be within 0.5 of this limit. Then you're telling me if this limit is actually true, you should be able to hand me a value around c. That if x is within that range, then f of x is definitely going to be as close to L as I desire. So let me draw that out to make it a little bit clearer. And I'm going to zoom in. I'm going to draw another diagram. So let's say that this right over here is my y-axis. And I'm going to zoom in. I'm going to draw a slightly different function, just so we can really focus on what's going on around here. The range is around c, and the range is around L. So that's x. This right over here is y. Let's say that this is c. And let's just zoom in on our function. So let's say our function looks, is doing something like, let's say it does something like, let's see, I don't want it to be defined at c. At least just for the-- it could be. You can always find a limit even where is defined. But let's say our function looks something like that. And it can have a little kink in it, the way I drew it. So it looks something like this. It's undefined. Let me draw it a little bit different. So it is undefined when x is equal to c. So this is the point where there's a hole. It is undefined when x is equal to c. So it even has a little kink in it, just like that. And what we want to do is prove that the limit, as x, the limit of f of x-- and let me make it clear, this is the graph of y is equal to f of x-- we want to get an idea for what this definition is saying. If we're claiming that the limit of f of x, as x approaches c, is L. So conceptually, we get the gist already. We already get the gist that this right over here is L. But what is this definition saying? Well, it's saying that you can get f of x as close to L as you want. So if you tell someone, I want to get f of x within a certain range of L, then if this limit is actually true, if the limit of f of x as x approaches c really is equal to L, then they should be able to find a range around c. That as long as x is around that range, your f of x is going to be in the range that you want. So let me actually go through that exercise. It really is a little bit like a game. So someone comes up to you and says, well, OK. I don't necessarily believe that you're claiming the limit of f of x as x approaches c is equal to L. I'm not really sure if that's the case. But I agree with this definition. So I want to get within 0.5. I want to get f of x within 0.5 of L. So this right over here would be L plus 0.5. And this right over here is L minus 0.5. And then you say, fine. I'm going to give you a range around c, that if you take any x within that range, your f of x is always going to fall in this range that you care about. And so you look at this-- and obviously we haven't explicitly defined this function. But you can even eyeball it, the way this function is defined. It won't be that easy for all functions. But you look at it like this. And you say that this value, just the way it's drawn right over here, let's say that this is c minus 0.25. And let's say that this value right over here is c plus 0.25. And so you tell them, look, as long as you get x within 0.25 of c, so as long as your x's are sitting someplace over here, the corresponding f of x is going to sit in the range that you care about. And you say, OK, fine. You won that round. Let me make it even tighter. Maybe instead of saying within the 0.5, I want to get within is 0.05. And then you'd have to do this exercise again and find another range. And in order for this to be true, you would have to be able to do this for any range that they give you. For any range around L that they give you, you have to be able to get f of x within that range by finding a range around c. That as long as x is that range around c, f of x is going to sit within that range. So I'll let you think about that a little bit. There's a lot to think about. But hopefully this made sense. We did it for the particular example of someone hands you the 0.5, I want f of x within the 0.5 of L, and you say, well, as long as x is within 0.25 of c, you're going to match it. You need to be able to do that for any range they give you around L. And then this limit will definitely be true. So in the next video, we will now generalize that. And that will really bring us to the famous epsilon delta definition of limits.