Trigonometric limits & squeeze theorem
Squeeze theorem intro
We're now going to think about one of my most favorite theorems in mathematics, and that's the squeeze theorem. And one of the reasons that it's one of my most favorite theorem in mathematics is that it has the word "squeeze" in it, a word that you don't see showing up in a lot of mathematics. But it is appropriately named. And this is oftentimes also called the sandwich theorem, which is also an appropriate name, as we'll see in a second. And since it can be called the sandwich theorem, let's first just think about an analogy to get the intuition behind the squeeze or the sandwich theorem. Let's say that there are three people. Let's say that there is Imran, let's say there's Diya, and let's say there is Sal. And let's say that Imran, on any given day, he always has the fewest amount of calories. And Sal, on any given day, always has the most number of calories. So in a given day, we can always say Diya eats at least as much as Imran. And then we can say Sal eats at least as much-- that's just to repeat those words-- as Diya. And so we could set up a little inequality here. On a given day, we could write that Imran's calories on a given day are going to be less than or equal to Diya's calories on that same day, which is going to be less than or equal to Sal's calories on that same day. Now let's say that it's Tuesday. Let's say on Tuesday you find out that Imran ate 1,500 calories. And on that same day, Sal also ate 1,500 calories. So based on this, how many calories must Diya have eaten that day? Well, she always eats at least as many as Imran's, so she ate 1,500 calories or more. But she always has less than or equal to the number of calories Sal eats. So it must be less than or equal to 1,500. Well, there's only one number that is greater than or equal to 1,500 and less than or equal to 1,500, and that is 1,500 calories. So Diya must have eaten 1,500 calories. This is common sense. Diya must have had 1,500 calories. And the squeeze theorem is essentially the mathematical version of this for functions. And you could even view this is Imran's calories as a function of the day, Sal's calories as a function of the day, and Diya's calories as a function of the day is always going to be in between those. So now let's make this a little bit more mathematical. So let me clear this out so we can have some space to do some math in. So let's say that we have the same analogy. So let's say that we have three functions. Let's say f of x over some interval is always less than or equal to g of x over that same interval, which is always less than or equal to h of x over that same interval. So let me depict this graphically. So that is my y-axis. This is my x-axis. And I'll just depict some interval in the x-axis right over here. So let's say h of x looks something like that. Let me make it more interesting. This is the x-axis. So let's say h of x looks something like this. So that's my h of x. Let's say f of x looks something like this. Maybe it does some interesting things, and then it comes in, and then it goes up like this, so f of x looks something like that. And then g of x, for any x-value, g of x is always in between these two. And I think you see where the squeeze is happening and where the sandwich is happening. If h of x and f of x were bendy pieces of bread, g of x would be the meat of the bread. So it would look something like this. Now, let's say that we know-- this is the analogous thing. On a particular day, Sal and Imran ate the same amount. Let's say for a particular x-value, the limit as f and h approach that x-value, they approach is the same limit. So let's take this x-value right over here. Let's say the x-value is c right over there. And let's say that the limit of f of x as x approaches c is equal to L. And let's say that the limit as x approaches c of h of x is also equal to L. So notice, as x approaches c, h of x approaches L. As x approaches c from either side, f of x approaches L. So these limits have to be defined. Actually, the functions don't have to be defined at x approaches c. Just over this interval, they have to be defined as we approach it. But over this interval, this has to be true. And if these limits right over here are defined and because we know that g of x is always sandwiched in between these two functions, therefore, on that day or for that x-value-- I should get out of that food-eating analogy-- this tells us if all of this is true over this interval, this tells us that the limit as x approaches c of g of x must also be equal to L. And once again, this is common sense. f of x is approaching L, h of x is approaching L, g of x is sandwiched in between it. So it also has to be approaching L. And you might say, well, this is common sense. Why is this useful? Well, as you'll see, this is useful for finding the limits of some wacky functions. If you can find a function that's always greater than it and a function that's always less than it, and you can find the limit as they approach some c, and it's the same limit, then you know that that wacky function in between is going to approach that same limit.