Squeeze Theorem

Intuition (but not a proof) of the Squeeze Theorem.

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.