[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]