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

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

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

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

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

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

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Then looking at my output values from my 
function, 

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right? 
So f(1) is sine of 1 / 1. 

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

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99.. 
A little bit more. 

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

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

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

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

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

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

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

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

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

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

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willing to have the input be close enough 
to 0. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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this. 
Let's take a look. 

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

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

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

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

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

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

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

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

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

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

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Just pictorially, why is this called the 
squeeze theorem? 

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I've got an example here. 
Three functions. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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approaches zero. 
And that means by the Squeeze Theorem, 

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