An older video where Sal introduces the squeeze (sandwich) theorem and its meaning. Created by Sal Khan.
In this video I will prove to you that the limit as x approaches 0 of sine of x over x is equal to 1. But before I do that, before I break into trigonometry, I'm going to go over another aspect of limits. And that's the squeeze theorem. Because once you understand what the squeeze theorem is, we can use the squeeze theorem to prove this. It's actually a pretty involved explanation, but I think you'll find it pretty neat and satisfying if you get it. If you don't get it, maybe you just want to memorize this. Because that's a very useful limit to know later on when we take the derivatives of trig functions. So what's the squeeze theorem? The squeeze theorem is my favorite theorem in mathematics, possibly because it has the word squeeze in it. Squeeze theorem. And when you read it in a calculus book it looks all complicated. I don't know when you read it, in a calculus book or in a precalculus book. It looks all complicated, but what it's saying is frankly pretty obvious. Let me give you an example. If I told you that I always-- so Sal always eats more than Umama. Umama is my wife. If I told you that this is true, Sal always eats more than Umama. And I were also to say that Sal always eats less than-- I don't know, let me make up a fictional character-- than Bill. So on any given day-- let's say this is in a given day. Sal always eats more than Umama in any given day, and Sal always eats less than Bill on any given day. Now if I were tell you that on Tuesday Umama ate 300 calories and on Tuesday Bill ate 300 calories. So my question to you is, how many calories did Sal eat, or did I eat, on Tuesday? Well I always eat more than Umama-- well, more than or equal to Umama-- and I always eat less than or equal to Bill. So then on Tuesday, I must have eaten 300 calories. So this is the gist of the squeeze theorem, and I'll do a little bit more formally. But it's essentially saying, if I'm always greater than one thing and I'm always less than another thing and at some point those two things are equal, well then I must be equal to whatever those two things are equal to. I've kind of been squeezed in between them. I'm always in between Umama and Bill, and if they're at the exact same point on Tuesday, then I must be at that point as well. Or at least I must approach it. So let me write it in math terms. So all it says is that, over some domain, if I say that, let's say that g of x is less than or equal to f of x, which is less than or equal to h of x over some domain. And we also know that the limit of g of x as x approaches a is equal to some limit, capital L, and we also know that the limit as x approaches a of h of x also equals L, then the squeeze theorem tells us-- and I'm not going to prove that right here, but it's good to just understand what the squeeze theorem is-- the squeeze theorem tells us then the limit as x approaches a of f of x must also be equal to L. And this is the same thing. This is example where f of x, this could be how much Sal eats in a day, this could be how much Umama eats in a day, this is Bill. So I always eat more than Umama or less than Bill. And then on Tuesday, you could say a is Tuesday, if Umama had 300 calories and Bill had 300 calories, then I also had to eat 300 calories. Let me let me graph that for you. Let me graph that, and I'll do it in a different color. Squeeze theorem. Squeeze theorem. OK, so let's draw the point a comma L. The point a comma L. Let's say this is a, that's the point that we care about. a, and this is L. And we know, g of x, that's the lower function, right? So let's say that this green thing right here, this is g of x. So this is my g of x. And we know that as g of x approaches-- so the g of x could look something like that, right? And we know that the limit as x approaches a of g of x is equal to L. So that's right there. So this is g of x. That's g of x. Let me do h of x in a different color. So now h of x could look something like this. Like that. So that's h of x. And we also know that the limit as x approaches a of h of x -- let's see, this is the function of x axis. So you can call it h of x, g of x, or f of x. That's just the dependent access, and this is the x-axis. So once again, the limit as x approaches a of h of x, well at that point right there, h of a is equal to L. Or at least the limit is equal to that. And none of these functions actually have to even be defined at a, as long as these limits, this limit exists and this limit exists. And that's also an important thing to keep in mind. So what does this tell us? f of x is always greater than this green function. It's always less than h of x, right? So any f of x I draw, it would have to be in between those two, right? So no matter how I draw it, if I were to draw a function, it's bounded by those two functions just by definition. So it has to go through that point. Or at least it has to approach that point. Maybe it's not defined at that point, but the limit as we approach a of f of x also has to be at point L. And maybe f of x doesn't have to be defined right there, but the limit as we approach it is going to be L. And hopefully that makes a little bit of sense, and hopefully my calories example made a little bit of sense to you. So let's keep that in the back of our mind, the squeeze theorem. And now we will use that to prove that the limit as x approaches 0 of sine of x over x is equal to 1. And I want to do that, one, because this is a super useful limit. And then the other thing is, sometimes you learn the squeeze theorem, you're like, oh, well that's obvious but when is it useful? And we'll see. Actually I'm going to do it in the next video, since we're already pushing 8 minutes. But we'll see in the next video that the squeeze theorem is tremendously useful when we're trying to prove this. I will see you in the next video.