Formal definition of limits (epsilon-delta)
Formal definition of limits Part 1: intuition review
Let's review our intuition of what a limit even is. So let me draw some axes here. So let's say this is my y-axis, so try to draw a vertical line. So that right over there is my y-axis. And then let's say this is my x-axis. I'll focus on the first quadrant, although I don't have to. So let's say this right over here is my x-axis. And then let me draw a function. So let's say my function looks something like that, could look like anything, but that seems suitable. So this is the function y is equal to f of x. And just for the sake of conceptual understanding, I'm going to say it's not defined at a point. I didn't have to do this. You can find the limit as x approaches a point where the function actually is defined, but it becomes that much more interesting, at least for me, or you start to understand why a limit might be relevant where a function is not defined at some point. So the way I've drawn it, this function is not defined when x is equal to c. Now, the way that we've thought about a limit is what does f of x approach as x approaches c? So let's think about that a little bit. When x is a reasonable bit lower than c, f of x, for our function that we just drew, is right over here. That's what f of x is going to be equal. y is equal to f of x. When x gets a little bit closer, then our f of x is right over there. When x gets even closer, maybe really almost at c, but not quite at c, then our f of x is right over here. And the way we see it, we see that our f of x seems to be-- as x gets closer and closer to c it looks like our f of x is getting closer and closer to some value right over there. Maybe I'll draw it with a more solid line. And that was actually only the case when x was getting closer to c from the left, from values of x less than c. But what happens as we get closer and closer to c from values of x that are larger than c? Well, when x is over here, f of x is right over here. And so that's what f of x is, is right over there. When x gets a little bit closer to c, our f of x is right over there. When x is just very slightly larger than c, then our f of x is right over there. And you see, once again, it seems to be approaching that same value. And we call that value, that value that f of x seems to be approaching as x approaches c, we call that value L, or the limit. And so the way we would denote it is we would call that the limit. We don't have to call it L all the time, but it is referred to as the limit. And the way that we would kind of write that mathematically is we would say the limit of f of x as x approaches c is equal to L. And this is a fine conceptual understanding of limits, and it really will take you pretty far, and you're ready to progress and start thinking about taking a lot of limits. But this isn't a very mathematically-rigorous definition of limits. And so this sets us up for the intuition. In the next few videos, we will introduce a mathematically-rigorous definition of limits that will allow us to do things like prove that the limit as x approaches c truly is equal to L.