AP®︎/College Calculus AB
Squeeze theorem intro
The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and using them to find the limit at x=0. Created by Sal Khan.
Want to join the conversation?
- I don't actually see the point of this. It's basically because there exist a possibility that f(x)=g(x)=h(x) there for there is one point that all 3 pass through. That's basically it. so is this what the squeeze theorem about?(20 votes)
- Yes on Khan Academy the squeeze theorem is straight forward. In a calc I class you will have to derive the two functions that will squeeze your original function. The squeeze theorem is used on a function where it will be merely impossible to differentiate. Therefore we will derive two functions that we know how to differentiate and we take the derivatives on those two functions at your specific point. Mind you one function has to be greater than or equal to the original function, and the other has to be smaller than or equal to the original function. When you take the derivatives on the two functions and you get the same answer, the original function (since it is between both functions) will also have to be that answer. Hope this helps!(87 votes)
- Around5:50Sal says that the functions don't need to be defined at c only when approaching c? How come this is the case? Don't the functions need to be defined at c as well to properly 'squeeze' the middle function?(45 votes)
- to continue from above, a limit is "getting infinitely close to but not actually dealing with the point".(9 votes)
- what if f(x) and h(x) touch each other on more than 1 point?-(17 votes)
- Then the squeeze theorem applies to all of those points.(30 votes)
- can squeeze theorem only applied on sin or it is applicable on cos and tangent(5 votes)
- You can use squeeze theorem on cosine.
For cosine, you would use -1 <= cosx <= 1 as your starting point.
On tangent, it would be -infinity < tanx < infinity.
If you have tangent, I'd suggest transforming the equation though, so it doesn't involve tangent, but has it in terms of sine and cosine, making it easier to work with.(7 votes)
- How come it's always f(x) or g(x) or h(x) when Sal(and about everybody else) talks about functions? Why can't there be some other names for functions? I never see i(x) or j(x).(2 votes)
- You certainly can use those letters as names for functions. It is just a common practice to start at f and go from there, but you don't have to.(7 votes)
- What will happen if at the point of all the graphs intersection.....both f(x) and h(x) have undefined value.....and have a discontinuity. Is squeeze theorem still applied?(3 votes)
- In general, all derivative operations require the function to be both continuous and differentiable. If either condition is violated, then any related or derived theorems can't be applied.(4 votes)
- Would this only be used when finding the limit of a trig function ?(2 votes)
- It can be used for other functions as well, such as polynomials.(5 votes)
- How can we compare functions, as like in the sandwich theorem f(x)<=g(x)<=h(x).How do we know that f(x) is always less than or equal to h(x)??(Assuming that we cannot draw the graph of the functions manually)(3 votes)
- It is assumed to be true as part of the axioms of the proof.(3 votes)
- does lim f(x) x->c=lim h(x) x->c basically mean f(x) and h(x) have a common point? and since g(x) is a point that has to be less than or equal h(x) but greater than or equal f(x) it would just be equal to the common point?is this a way to think about this?(3 votes)
- Yes, the common point is
(c, g(c)). But just as Sal says at5:55, the three functions don't necessarily have to be defined at x = c; only that their limit has to be the same ie., all the three functions tend to the same value (L) at x = c. So this 'common point' might not even be defined.(3 votes)
- Just a tiny doubt: do the inequalities have to be slack inequalities (≤ or ≥) or would strict inequalities (< or >) work as well?(3 votes)
- What do you think?
The answer is yes, it has to be less or equal. This is because sin and cos have values between 1 and -1, with this values included. Depending on the trigonometric function you are working with, you will have different bounds and you will have to use less or equal/greater or equal or less/greater.(2 votes)
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.