Name: Limit of sin(x)/x as x approaches 0 | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T07:38:28.000Z
Duration: 9 min 16 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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

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,

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

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

Can I express that in terms of a trigonometric function?

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

The adjacent side down here, 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

in both the first and the fourth quadrant

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

that sits in this wedge, in this pie piece,

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

which is the absolute value of the sine of theta.

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,

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

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

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,

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,

which is the absolute value of tangent of theta.

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

of this pink or this salmon-colored triangle

and how do you compare that area of the wedge

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

is less than or equal to the area of the wedge

Subtitles ListPlay Video

Limit of sin(x)/x as x approaches 0 | Derivative rules | AP Calculus AB | Khan Academy

entire

deserve

positive

negative