Squeeze theorem intro

we're now going to think about one of my most favorite theorems and mathematics and that's the squeeze theorem and one of the reasons that it's one of my most favorite theorems in mathematics is that it has the word squeeze in it or 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 for to kind of 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 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 he always has the fewest amount of calories and Sal on any given day always has the most number of calories so on a given day on a given day we can always say the eats at least as much at least as much much as Imran and then we can say Sal Sal eats at least as much that 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 in Ron's calories in Ron's calories on a given day are going to be less than or equal to D as calories D is calories on that same day which is going to be less than or equal to Sal's calories on that same day Sal's calories on that same day now let's say that it's Tuesday let's say on Tuesday you find out that M Ron Emraan ate 1500 calories 1500 calories and on that same day Sal also eight Sal also eight 1500 calories so based on this how many calories must diya have eaten that day well she ought she always eats at least as many Zimmer ons so she ate 1500 calories or more and but she always has less than the numb or equal to the number of calories Sal eats so it must be less than or equal to 1500 well there's only one number that is greater than or equal to 1500 and less than or equal to 1500 and that is 1500 calories so the must have eaten 1500 calories this is common sense dia dia must have had 1500 calories and the squeeze theorem is essentially the mathematical version of this for functions and you could even view this is Emraan calories as a function of the day Sal's calories as a function of the day and these 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 a mathematical so let's 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 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 also always less than or equal to H of x over that same interval so let me depict this graphically so let's depict it graph graphically right over here so that is my y-axis this is my x-axis this is my x-axis and I'll just depict some interval on the x-axis right over here so let's say H of X looks something like H of X looks something like that to make it more interesting whoops this is 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 look something like this maybe it does some interesting things and then it comes in 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 so G of X is always in between this and I think you see where the squeeze is happening and where the sandwich is happening so this looks like a 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 analogous sitting 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 approach that x value they approach the same limit so let's sake this x value right over here let's say X value is C right over there and let's say that the limit the limit of f of X as X approaches C as X approaches C is equal to is equal to L is equal to L and let's say that the limit as X approaches C of H of X of H of X is also is also equal to L so those as X approaches C H of X approaches L as X approaches C from either sides f of X approaches L so these limits have to be defined that 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 because we know that G of X is always sandwiched in between these two functions therefore on that day or that for that x value I should get out of that food-eating analogy this tells us this tells us if all of this is true over this interval this tells us that the limit as X the limit as X approaches C of G of X of G of X must also be equal to L and once again this is common hence f of X is approaching out H of X is approaching out G of X is sandwiched in between it so it also has to be it also has to be approaching out 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 whacky functions if you can find a function that's always greater than it and a function that's always less than and you can find the limit as they approach some see it's the same limit then you know that that wacky function in between is going to approach that same limit