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• - [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

• 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