Placeholder Image

字幕列表 影片播放

  • - [Voiceover] Let's see if we can find the limit

  • as theta approaches zero of one minus cosine theta

  • over two sine squared theta.

  • And like always, pause the video and see if you

  • could work through this.

  • Alright, well our first temptation is to say,

  • "Well, this is going to be the same thing

  • "as the limit of one minus cosine theta

  • "as X approaches, or not X,

  • "as theta approaches zero.

  • "of theta, as theta approaches zero,

  • "over the limit,

  • "as theta approaches zero

  • "of two sine squared theta."

  • Now, both of these expressions

  • which could be used to define a function,

  • that they'd be continuous if you graph them.

  • They'd be continuous at theta equals zero,

  • so the limit is going to be the same thing,

  • as just evaluating them at theta equals zero

  • So this is going to be equal to one minus cosine of zero

  • over two sine squared of zero.

  • Now, cosine of zero is one and then one minus one is zero,

  • and sine of zero is zero, and you square it,

  • You still got zero and you multiply times two,

  • you still got zero.

  • So you got zero over zero.

  • So once again, we have that indeterminate form.

  • And once again, this indeterminate form

  • when you have zero over zero, doesn't mean to give up,

  • it doesn't mean that the limit doesn't exist.

  • It just means, well maybe

  • there's some other approaches here to work on.

  • If you got some non-zero number divided by zero,

  • then you say, okay that limit doesn't exist

  • and you would say, well you just say it doesn't exist.

  • But let's see what we can do to maybe, to maybe think

  • about this expression in a different way.

  • So if we said,

  • so let's just say that this,

  • let me use some other colors here.

  • Let's say that this right over here

  • is F of X.

  • So, F of X is equal to one minus cosine theta

  • over two sine squared theta,

  • and let's see if we can rewrite it in some way

  • that at least the limit as theta approaches zero

  • isn't going to, we're not gonna get the same

  • zero over zero.

  • Well, we can, we got some trig functions here,

  • so maybe we can use some of our trig identities

  • to simplify this.

  • And the one that jumps out at me

  • is that we have the sine squared of theta

  • and we know from the Pythagorean, Pythagorean Identity

  • in Trigonometry, it comes straight out of the unit circle

  • definition of sine and cosine.

  • We know that, we know that sine squared theta

  • plus cosine squared theta is equal to one

  • or, we know that sine squared theta

  • is one minus cosine squared theta.

  • One minus cosine squared theta.

  • So we could rewrite this.

  • This is equal to one minus cosine theta

  • over two times one minus cosine squared theta.

  • Now, this is one minus cosine theta.

  • This is a one minus cosine squared theta,

  • so it's not completely obvious yet

  • of how you can simplify it,

  • until you realize that this could be viewed

  • as a difference of squares.

  • If you view this as,

  • if you view this as A squared minus B squared,

  • we know that this can be factored as A plus B

  • times A minus B.

  • So I could rewrite this.

  • This is equal to one minus cosine theta

  • over two times,

  • I could write this as one plus cosine theta

  • times one minus cosine theta.

  • One plus cosine theta times one minus, one minus

  • cosine theta.

  • And now this is interesting.

  • I have one minus cosine theta

  • in the numerator and I have a one minus cosine theta

  • in the denominator.

  • Now we might be tempted to say,

  • "Oh, let's just cross that out with that

  • "and we would get, we would simplify it

  • "and get F of X is equal to one over

  • "and we could distribute this two now."

  • We could say, "Two plus two cosine theta."

  • We could say,

  • "Well, aren't these the same thing?"

  • And we would be almost right,

  • because F of X, this one right over here,

  • this, this is defined

  • this right over here is defined

  • when theta is equal to zero,

  • while this one is not defined when theta

  • is equal to zero.

  • When theta is equal to zero,

  • you have a zero in the denominator.

  • And so what we need to do in order

  • for this F of X or in order

  • to be, for this to be the same thing,

  • we have to say, theta cannot be equal to zero.

  • But now let's think about the limit again.

  • Essentially, what we want to do is we want to find

  • the limit as theta approaches zero

  • of F of X.

  • And we can't just do direct substitution

  • into, if we do, if we really take this seriously,

  • 'cause we're gonna like,

  • "Oh well, if I try to put zero here,

  • "it says theta cannot be equal to zero

  • "F of X is not defined at zero."

  • This expression is defined at zero

  • but this tells me,

  • "Well, I really shouldn't apply zero to this function."

  • But we know that if we can find another function

  • that is defined, that is the exact same thing as

  • F of X except at zero,

  • and it is continuous at zero.

  • And so we could say,

  • "G of X is equal to one over two plus two

  • "cosine theta."

  • Well then we know this limit is going to be the

  • exact same thing as the limit

  • of G of X, as theta approaches zero.

  • Once again, these two functions

  • are identical except F of X is not defined

  • at theta equals zero,

  • while G of X is.

  • But the limits as theta approaches zero

  • are going to be the same.

  • And we've seen that in previous videos.

  • And I know what a lot of you are thinking.

  • Sal, this seems like a very, you know,

  • why don't I just, you know, do this algebra here.

  • Cross these things out of this.

  • Substitute zero for theta.

  • Well you could do that and you would get the answer,

  • but you need to be clear if, or it's important

  • to be mathematically clear of what you are doing.

  • If you do that, if you just crossed these two out

  • and all of a sudden you're expression

  • becomes defined at zero,

  • you are now dealing with a different expression

  • or a different function definition.

  • So to be clear, if you want to say this is the function

  • you're finding the limit of,

  • you have to put this constraint in

  • to make sure it has the exact same domain.

  • But lucky for us, we can say,

  • if we've had another, another function that's continuous

  • at that point that doesn't have that gap there,

  • that doesn't have that point discontinuity out,

  • the limits are going to be equivalent.

  • So the limit as theta approaches zero of G of X,

  • well, that's just going to be

  • since it's continuous at zero.

  • We could say that's just going to be,

  • we can just substitute.

  • That's going to be equal to G of zero

  • which is equal to one over two plus

  • cosine two, one over two plus two times cosine

  • of zero.

  • Cosine of zero is one,

  • so it's just one over two plus two,

  • which is equal to,

  • deserve a little bit of a drum roll here.

  • Which is equal to one fourth.

  • And we are done.

- [Voiceover] Let's see if we can find the limit