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Course: Calculus, all content (2017 edition)>Unit 1

Lesson 14: Squeeze theorem

Squeeze theorem intro (old)

An older video where Sal introduces the squeeze (sandwich) theorem and its meaning. Created by Sal Khan.

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• Couldn't we just use L'hopital's rule:
(d/dx)sin x->cos x->1
(d/dx)x ->1
• This video is meant to be seen before derivatives, hence no L'hopital.
• Why do they call it sqeeze theorem?-Selah Bryce
• if something is between two objects, such as a letter between A and G, you can narrow it down to which letter if you "squeeze" it down to, say, between D and F
Squeezing is another name for narrowing it down and zeroing in on the correct answer.
• Is this also called the Sandwich Theorem? I vaguely remember this from last year...
• using the squeeze theorem how can I evaluate: lim x-> infinity e^-8x cos x

cos x will always be bound between -1 and 1 so do I set it up so do I set it up where

-e^-8x</= e^-8x cos x</= e^-8x?
• You really don't need to. As the limit of e^-8x as x approaches infinity is 0 already, multiplying anything will give 0. However, if you must, your function is correct. Because the limit of both of those approaches 0, the function's amplitude becomes less and less exponentially until it approaches no variation because of the squeeze that is forced by -e^-8x and e^-8x coming together, restricting from both above and below.
• Just checking to be sure, the lines on the graph were just lines that intersect at (L,a) and their shape was nothing more than randomly drawn, correct?
(1 vote)
• the functions squeezing f were randomly drawn, though they weren't lines.
• I was trying to test the very first equation that says the limit as x approaches 0 for sinx/x = 1, but I am getting a different number than 1 for the limit. When I use a number like 0.0000000001 for x, I get 0.017453292. Shouldn't the answer be 1? Am I doing something wrong? Do I not understand limits? Or has Sal made a mistake?
(1 vote)
• You made a mistake somewhere in your calculation.
Let's look at an easier value for x:
x = 0.1 sin(0.1) = 0.0998 so sin (0.1)/0.1 = 0.998, which is pretty close to 1.
• If it was less than and more than, then wouldn't it not be true because they are equal and f(x) could be free! Right?
(1 vote)
• Remember, we are computing the limit, not the actual value of f(x) if h(x) < f(x) < g(x). Although by this logic, h(x) can't equal g(x), the limit as x approaches c of h(x) might be equal to the limit as x approaches c of g(x).
• Can spmeone help find the limit and confirm using the sandwich theorem

Lim(1-cosx)/(x^2)
X->infinity
(1 vote)
• Since the numerator is at least 0 and at most 2, then it follows that. (Or, stated more formally, the range of the numerator is from [0 to 2] inclusive}.
0/x² ≤ (1-cosx)/(x²) ≤ 2/x²
Since lim x→ ∞ 0/x² = 0
And since lim x→ ∞ 2/x² = 0
Then, sandwich requires that lim x→ ∞ (1-cos x)/(x²) = 0
(1 vote)
• help me to solve this: if limit as x approaches 8 of 1 over (x-8) equals to infinity, how about limit as x approaches 5 of 1 over (x-5)?
(1 vote)
• No Lim x->8 1/(x-8) is undefined. This is because when when you approach from left you get negative infinity, and when you approach from the right you get positive infinity. This is true for all problems similar to this.
(1 vote)
• Do most "Squeeze Theorem" have an "greater than or equal to"/" Less than or equal to" ratio?
(1 vote)
• There has to be a known relationship between the limit values of the three functions, because only by using this relationship can we determine the value of the 'center' function. It would probably also be possible to apply the Squeeze Theorem to a situation where a = b or b = c, but in that case we would already know the value, so the Squeeze Theorem would not really need to be employed.
(1 vote)

Video transcript

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.