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University of California - Los Angeles

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

University of California - Los Angeles

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