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Current time:0:00Total duration:9:16

AP Calc: LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.2 (EK)

- [Instructor] What we're
going to do in this video 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.