WEBVTT
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All right, So in this question, what we
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need to do is find the limit of the function
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of effects, which is full attention for X over
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contention three X His ex approaches zero. So first
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thing we do is we write that down in the
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hopper to notation just the limit as export zero of
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co tangent of for X over. So tensions three
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x and this will give us the limit and sex
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approaches zero using the trigger Ministry identities of co sign
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for X. Well, we're sign of for X
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over the side of three acts over. Sign of
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reacts weigh right. Everything as limit as X approaches
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. Zero of co sign for X sign of three
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X over. No sign of reacts. Sign of
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four X. Okay, So the trick to solving
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this problem Fine. We had this one nasty term
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down here, which is throwing us off. So
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the trick is solving n'est promise to remember that the
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limit of Moses actually used something different than X.
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The limit as Fada approaches zero. So sign Fada
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over data is equal to one. And similarly,
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the limit has Athena approaches zero of Fada who our
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sign beta is equal to the limit s data approaches
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zero of one over. Sign data over Nina.
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I'LL let you verify this for yourself, and we
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could just break these down into their respective limits.
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So the limit of one estate approaches zero is one
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and the limit of state approaches. Zero of scientist
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overthe data, as we've previously found out here is
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just one. So when so we can use these
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two instruments is too tricks that we have to rewrite
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this problem and let this solve for this limit now
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, Okay, so let's rewrite everything again. So
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we say that the limits of co sign for X
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slover coastline reacts. That's Tex approaches zero times the
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limit. The sign three acts over for X books
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for sign of for X No sex approaches and Sarah
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were when we're going to do this because lim allow
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us to break down the firm's we know this this
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first limb is actually pretty easy. Assist co sign
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for X over coast and three X as co sign
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before X approaches. Zero. It's just cause and
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zero over zero. Just this one. So one
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times Now, this is the tricky part. What
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we need to do here is multiply both by four
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X and also by three ex. Okay, so
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I'll show you how that works here. So as
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limits Thanks approaches zero of four x Nice three x
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for X times three Experian effect just multiplying by one
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. Right. So this is This is that's what
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permits us to do this A nifty little trick of
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sign. Three acts over, sign for X.
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Right? So now, looking back and this identity
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that we have here or that this limit law that
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we have here right where science fate over theta equals
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one as limit the fate of ghost zero, we
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can apply and out rule right here. The way
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we do that is we can group these terms,
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right? We know that. Yeah. The slips
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you have this three x signed reacts. We also
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have the sign for X. All of this for
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X, right? So what's again? Let's rewrite
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this one times. Well, we got this one
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from this limit up here, So one times the
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limit has X approaches. Zero ofthree ax over for
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X times the limit. A sex approaches zero no
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for X. Over sign of for x times the
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limits The sex approaches zero of science three X clover
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. Three acts okay, And remember, right using
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these fleming laws that we have here Well, we
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get is limit of for X over. Sign off
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for access X approaches zero Well, we know that
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one three x sign of freaks over three Xs explosions
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there. We know that's one. And now that
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we're left with is this limit of X as export
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is a row of three X over for X,
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that's easy. Just cancel off his access. Then
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we know this guy's equal tio three fourths. So
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the limit of this whole thing of contended for X
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over coach engine three exits just three over four,
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right one times three or four times one times one
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agains us three over for