[MUSIC] So here's a great function to 
look at. 
The function is going to be defined by 
f(x) = sine x / x. 
My question is what's the limit of this 
function, as x approaches zero. 
Let's try to guess the limit by looking 
at a table of function values. 
So here's a bunch of input values that 
are getting closer and closer to zero, 
right.1, .01, .001. 
It's getting closer and closer to zero. 
Then looking at my output values from my 
function, 
right? 
So f(1) is sine of 1 / 1. 
It's the sine of 1. 
f(.1), that's sign of 1. over 1,. it's 
99.. 
A little bit more. 
f(.01) is 9999. a little bit more. 
And you keep looking down here and these 
numbers seem to be getting close to 
something, alright. 999999. this is 
really, really close to one. 
So based on this table of values your 
tempted to guess that the limit of f(x) 
as x approaches 0 is 1. 
Another way to gain some insight about 
this limit will be to look at the graph. 
Here's the graph. 
This is the graph of sign x over x. 
And you can see that when x equals zero, 
functions not defined there because I 
can't divide by zero, so I got this 
little hole in the graph. 
Nevertheless, I'm claiming that the limit 
as x approaches 0 is equal to 1 which 
actually means that I can make the output 
as close to 1 as you like, if you're 
willing to have the input be close enough 
to 0. 
Instead of talking about closeness, push 
this red button and turn on this red 
interval. 
So when I say close to one, what I really 
mean is the output is inside this, this 
red interval. 
And that red interval might be really big 
or it might be really small. 
But to be close to one is going to mean 
inside the red interval. 
The point is that, can turn on this blue 
interval. 
And as close as you want the output to be 
the one, I can promise you that the 
output is within the red interval if the 
input is within this blue interval. 
When the red interval is really big, well 
that's not much of a challenge. 
I can have a really wide blue interval 
and anything inside the blue interval has 
output landing inside the red interval. 
But even when the red interval is very, 
very small there's still some tiny blue 
interval so that whenever x is within the 
blue interval, the output is within the 
tiny red interval. 
In other words, even if you want the 
output to be really close to one. 
I can promise you that the output is that 
close to one, if you're willing to have 
the input be close enough to zero. 
So, we've looked at the function values, 
we've looked at the graph. 
We've got this idea that the limit of 
sine x over x as x approaches zero is 
equal to one. 
But it's just that, it's just an idea. 
We don't yet have a rigorous argument 
that this limit is equal to one. 
Here's a sketch of a more rigorous 
argument that the limit of sine x / x, as 
x approaches 0 is equal to one. 
It turns out that for values of x which 
are close to but not equal to zero, this 
is true. 
Cosine of x is less than sine x over x, 
and sine x over x is less than one. 
Now why would you care about this? 
Note, the limit of cosine x as x 
approaches zero is one and the limit of 1 
is 1 because the limit of a constant 
function is just that constant. 
So I know that the limit of this side is 
one and the limit of this side is one and 
what I'm trying to conclude is that the 
limit of the thing in between is also 
one. 
And it turns out there's a way to do 
this. 
Let's take a look. 
Here's what we're going to use, the 
squeeze theorum. 
Suppose you've got three functions, I'm 
calling them GF and H. 
G(x) is less than equal to f(x) and f(x) 
is less than equal to H(x). 
For values of x that are near A, but 
maybe these inner qualities don't hold at 
the point A. 
Also, suppose that the limit of G(x) as x 
approaches A, is equal to the limit of 
H(x) as x approach A, is equal to sum L. 
So the limit of G(x), the limit of H of X 
are the same value, L. 
The, you get to conclude the limit of f 
as x approaches a exists and it equals l. 
Why is this thing called the Squeeze 
Theorem or some people call it the 
Sandwich Theorem or the Pinching Theorem? 
Let's take a look. 
Just pictorially, why is this called the 
squeeze theorem? 
I've got an example here. 
Three functions. 
G, F, and H. 
And again, G(x) is less than F(x), F(x) 
less than H(x). 
Now, note, the limit of G(x) as x 
approaches A is L. 
And the limit of H(x) as x approaches A 
is L. 
F is squeezed, or sandwiched, between H 
and G. 
And consequently, the limit of f as x 
approaches A is also equal to L. 
Now, we're going to use the squeeze 
theorem to try to understand the limit of 
sin x over x. 
So we've got the Squeeze Theorem. 
And what do I know? 
I know that cosine X is less than sine x 
/ x is less than 1 for values of x that 
are close to but not equal to 0. 
And the limit of cosine x as x approaches 
0 is equal to 1. 
If you like, because cosines continuous 
and cosine of 0 is 1. 
Also the limit of 1 as x approaches 0 is 
equal to 1 because the limit of a 
constant function is that constant. 
So the limit of this function is one, the 
limit of this function is one as x 
approaches zero. 
And that means by the Squeeze Theorem, 
the limit of sine x / x is also equal to 
1 [MUSIC]