is prove that the limit

as theta approaches zero

of sine of theta

over theta is equal to one.

So let's start with a little bit of a geometric

or trigonometric construction that I have here.

So this white circle, this is a unit circle,

that we'll label it as such.

So it has radius one,

unit

circle.

So what does the length

of this salmon-colored line represent?

Well, the height of this line would be the y-coordinate

of where this radius intersects the unit circle.

And so by definition, by the unit circle definition

of trig functions, the length of this line

is going to be sine of theta.

If we wanted to make sure that also worked for thetas

that end up in the fourth quadrant, which will be useful,

we can just insure that it's the absolute value

of the sine of theta.

Now what about this blue line over here?

Can I express that in terms of a trigonometric function?

Well, let's think about it.

What would tangent of theta be?

Let me write it over here.

Tangent of theta

is equal to opposite over adjacent.

So if we look at this broader triangle right over here,

this is our angle theta in radians.

This is the opposite side.

The adjacent side down here, this just has length one.

Remember, this is a unit circle.

So this just has length one,

so the tangent of theta is the opposite side.

The opposite side is equal to the tangent of theta.

And just like before, this is going to be a positive value

for sitting here in the first quadrant

but I want things to work

in both the first and the fourth quadrant

for the sake of our proof,

so I'm just gonna put an absolute value here.

So now that we've done,

I'm gonna think about some triangles

and their respective areas.

So first, I'm gonna draw a triangle

that sits in this wedge, in this pie piece,

this pie slice within the circle,

so I can construct this triangle.

And so let's think about the area

of what I am shading in right over here.

How can I express that area?

Well, it's a triangle.

We know that the area of a triangle

is 1/2 base times height.

We know the height

is the absolute value of the sine of theta

and we know that the base is equal to one,

so the area here is going to be equal to 1/2

times our base, which is one,

times our height,

which is the absolute value of the sine of theta.

I'll rewrite it over here.

I can just rewrite that

as the absolute value of the sine of theta over two.

Now let's think about the area of this wedge

that I am highlighting in this yellow color.

So what fraction of the entire circle is this going to be?

If I were to go all the way around the circle,

it would be two pi radians,

so this is theta over to two pis of the entire circle

and we know the area of the circle.

This is a unit circle, it has a radius one,

so it'd be times the area of the circle,

which would be pi times the radius square,

the radius is one, so it's just gonna be times pi.

And so the area of this wedge right over here,

theta over two.

And if we wanted to make this work

for thetas in the fourth quadrant,

we could just write an absolute value sign right over there

'cause we're talking about positive area.

And now let's think about this larger triangle

in this blue color, and this is pretty straightforward.

The area here is gonna be 1/2 times base times height.

So the area, and once again, this is this entire are,

that's going to be 1/2 times our base, which is one,

times our height,

which is the absolute value of tangent of theta.

And so I can just write that down

as the absolute value of the tangent of theta over two.

Now, how would you compare the areas

of this pink or this salmon-colored triangle

which sits inside of this wedge

and how do you compare that area of the wedge

to the bigger triangle?

Well, it's clear that the area of the salmon triangle

is less than or equal to the area of the wedge

and the area of the wedge is less than or equal to

the area of the big, blue triangle.

The wedge includes the salmon triangle

plus this area right over here,

and then the blue triangle includes the wedge

plus it has this area right over here.

So I think we can feel good visually

that this statement right over here is true

and I'm just gonna do

a little bit of algebraic manipulation.

Let me multiply everything by two

so I can rewrite that the absolute value

of sine of theta is less than or equal to

the absolute value of theta

which is less than or equal to the absolute value

of tangent of theta, and let's see.

Actually, instead of writing the absolute value

of tangent of theta, I'm gonna rewrite that

as the absolute value of sine of theta

over the absolute value of cosine of theta.

That's gonna be the same thing

as the absolute value of tangent of theta.

And the reason why I did that

is we can now divide everything

by the absolute value of sine of theta.

Since we're dividing by a positive quantity,

it's not going to change the direction of the inequalities.

So let's do that

I'm gonna divide this

by an absolute value of sine of theta.

I'm gonna divide this

by an absolute value of the sine of theta

and then I'm gonna divide this

by an absolute value of the sine of theta.

And what do I get?

Well, over here, I get a one

and on the right-hand side, I get a one

over the absolute value of cosine theta.

These two cancel out.

So the next step I'm gonna do

is take the reciprocal of everything.

And so when I take the reciprocal of everything,

that actually will switch the inequalities.

The reciprocal of one is still going to be one

but now, since I'm taking the reciprocal of this here,

it's gonna be greater than or equal to

the absolute value of the sine of theta

over the absolute value of theta,

and that's going to be greater than or equal to

the reciprocal of one

over the absolute value of cosine of theta

is the absolute value of cosine of theta.

We really just care about the first and fourth quadrants.

You can think about this theta

approaching zero from that direction

or from that direction there,

so that would be the first and fourth quadrants.

So if we're in the first quadrant and theta is positive,

sine of theta is gonna be positive as well.

And if we're in the fourth quadrant and theta's negative,

well, sine of theta is gonna have the same sign.

It's going to be negative as well.

And so these absolute value signs aren't necessary.

In the first quadrant,

sine of theta and theta are both positive.

In the fourth quadrant, they're both negative,

but when you divide them,

you're going to get a positive value, so I can erase those.

If we're in the first or fourth quadrant,

our X value is not negative,

and so cosine of theta, which is the x-coordinate

on our unit circle, is not going to be negative,

and so we don't need the absolute value signs over there.

Now, we should pause a second

because we're actually almost done.

We have just set up three functions.

You could think of this as f of x is equal to,

you could view this as f of theta is equal to one,

g of theta is equal to this,

and h of theta is equal to that.

And over the interval that we care about,

we could say for negative pi over two

is less than theta is less than pi over two,

but over this interval, this is true for any theta

over which these functions are defined.

Sine of theta over theta is defined over this interval,

except where theta is equal to zero.

But since we're defined everywhere else,

we can now find the limit.

So what we can say is, well, by the squeeze theorem

or by the sandwich theorem,

if this is true over the interval,

then we also know that the following is true.

And this, we deserve a little bit of a drum roll.

The limit

as theta approaches zero of this

is going to be greater than or equal to the limit

as theta approaches zero of this,

which is the one that we care about,

sine of theta over theta,

which is going to be greater than or equal to the limit

as theta approaches zero of this.

Now this is clearly going to be just equal to one.

This is what we care about.

And this, what's the limit as theta approaches zero

of cosine of theta?

Well, cosine of zero is just one

and it's a continuous function,

so this is just gonna be one.

So let's see.

This limit is going to be less than or equal to one

and it's gonna be greater than or equal to one,

so this must be equal to one

and we are done.