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[MUSIC] So here's a great function to look at.

Â The function is going to be defined by f(x) = sine x / x.

Â My question is what's the limit of this function, as x approaches zero.

Â Let's try to guess the limit by looking at a table of function values.

Â So here's a bunch of input values that are getting closer and closer to zero,

Â right.1, .01, .001. It's getting closer and closer to zero.

Â Then looking at my output values from my function,

Â right? So f(1) is sine of 1 / 1.

Â It's the sine of 1. f(.1), that's sign of 1. over 1,. it's

Â 99.. A little bit more.

Â f(.01) is 9999. a little bit more. And you keep looking down here and these

Â numbers seem to be getting close to something, alright. 999999. this is

Â really, really close to one. So based on this table of values your

Â tempted to guess that the limit of f(x) as x approaches 0 is 1.

Â Another way to gain some insight about this limit will be to look at the graph.

Â Here's the graph. This is the graph of sign x over x.

Â And you can see that when x equals zero, functions not defined there because I

Â can't divide by zero, so I got this little hole in the graph.

Â Nevertheless, I'm claiming that the limit as x approaches 0 is equal to 1 which

Â actually means that I can make the output as close to 1 as you like, if you're

Â willing to have the input be close enough to 0.

Â Instead of talking about closeness, push this red button and turn on this red

Â interval. So when I say close to one, what I really

Â mean is the output is inside this, this red interval.

Â And that red interval might be really big or it might be really small.

Â But to be close to one is going to mean inside the red interval.

Â The point is that, can turn on this blue interval.

Â And as close as you want the output to be the one, I can promise you that the

Â output is within the red interval if the input is within this blue interval.

Â When the red interval is really big, well that's not much of a challenge.

Â I can have a really wide blue interval and anything inside the blue interval has

Â output landing inside the red interval. But even when the red interval is very,

Â very small there's still some tiny blue interval so that whenever x is within the

Â blue interval, the output is within the tiny red interval.

Â In other words, even if you want the output to be really close to one.

Â I can promise you that the output is that close to one, if you're willing to have

Â the input be close enough to zero. So, we've looked at the function values,

Â we've looked at the graph. We've got this idea that the limit of

Â sine x over x as x approaches zero is equal to one.

Â But it's just that, it's just an idea. We don't yet have a rigorous argument

Â that this limit is equal to one. Here's a sketch of a more rigorous

Â argument that the limit of sine x / x, as x approaches 0 is equal to one.

Â It turns out that for values of x which are close to but not equal to zero, this

Â is true. Cosine of x is less than sine x over x,

Â and sine x over x is less than one. Now why would you care about this?

Â Note, the limit of cosine x as x approaches zero is one and the limit of 1

Â is 1 because the limit of a constant function is just that constant.

Â So I know that the limit of this side is one and the limit of this side is one and

Â what I'm trying to conclude is that the limit of the thing in between is also

Â one. And it turns out there's a way to do

Â this. Let's take a look.

Â Here's what we're going to use, the squeeze theorum.

Â Suppose you've got three functions, I'm calling them GF and H.

Â G(x) is less than equal to f(x) and f(x) is less than equal to H(x).

Â For values of x that are near A, but maybe these inner qualities don't hold at

Â the point A. Also, suppose that the limit of G(x) as x

Â approaches A, is equal to the limit of H(x) as x approach A, is equal to sum L.

Â So the limit of G(x), the limit of H of X are the same value, L.

Â The, you get to conclude the limit of f as x approaches a exists and it equals l.

Â 4:32

Why is this thing called the Squeeze Theorem or some people call it the

Â Sandwich Theorem or the Pinching Theorem? Let's take a look.

Â Just pictorially, why is this called the squeeze theorem?

Â I've got an example here. Three functions.

Â G, F, and H. And again, G(x) is less than F(x), F(x)

Â less than H(x). Now, note, the limit of G(x) as x

Â approaches A is L. And the limit of H(x) as x approaches A

Â is L. F is squeezed, or sandwiched, between H

Â and G. And consequently, the limit of f as x

Â approaches A is also equal to L. Now, we're going to use the squeeze

Â theorem to try to understand the limit of sin x over x.

Â So we've got the Squeeze Theorem. And what do I know?

Â I know that cosine X is less than sine x / x is less than 1 for values of x that

Â are close to but not equal to 0. And the limit of cosine x as x approaches

Â 0 is equal to 1. If you like, because cosines continuous

Â and cosine of 0 is 1. Also the limit of 1 as x approaches 0 is

Â equal to 1 because the limit of a constant function is that constant.

Â So the limit of this function is one, the limit of this function is one as x

Â approaches zero. And that means by the Squeeze Theorem,

Â the limit of sine x / x is also equal to 1 [MUSIC]

Â