find-the-limit-displaystyle-lim_theta-to-0-frac-sin-thetatheta-tan-theta

Question

Find the limit.
$ \displaystyle \lim_{	heta 	o 0} \frac {\sin 	heta}{	heta + 	an 	heta} $

Frank Lin · Accepted Answer

And for this problem, we use the fact that science it on sitar tendency. That&#x27;s in our simple optical equipment in Sedona course zero. So we can replace scientist and attendance it up. I said that when we take it only takes a limit, so this limit will just be one over two.

Zachary Mitchell · Answer

We&#x27;re starting with limit as X goes to zero of tan of Excuse me, I&#x27;ve written x here, but I meant fate. Oh, 10 of data divided by Fada. Now, let&#x27;s go ahead and check to see if we can use low petals rule, um Or if we can just do this limit. So when we try to do the limited the numerator, we get zero try to do the limited. The denominator would get 0 0/0 We have an indeterminant form and we can use lobaton. Okay, some reading LH over the equal sign indicating that that&#x27;s the next step low Patel&#x27;s room. So the numerator we get, she can&#x27;t squared because the derivative of Tanis against weird the Trudeau of data is just one. And this is something we can evaluate. Lemonis one

Amy Jiang · Answer

forget the rest on the limits, as they approach is, they&#x27;re all of sign. There are over. Sign to data. Okay, we can rewrite this as actually, it&#x27;s multiplied by two later. And divide by data and divide by that as well. So we get the limits instead of approaches, uh, sighing data we&#x27;re gonna divide by data and also multiplied by data. Let&#x27;s do that. And then we&#x27;re gonna divide by two data and also multiply right to data all over sign, too. Okay. And then now plugging in zero arching limit as they&#x27;re ever. So we know that this is eagle one. This is also equal to one, and then we just gets used to terrify, and we&#x27;re left with one hand.

Issa Dababneh · Answer

So in this question, where as to determine the limited data goes to zero off sign Dana divided by sine two things. And we have to use the intimate relationship that has that limit. Its K goes to zero scientific one, and then directly. What it buys but we can get for using laws off limits is that the Limited&#x27;s cake goes to zero k, divided by sine K is equal to one, so their reciprocal is also called toe the limit of the reciprocal. So the first thing we&#x27;re gonna do is we&#x27;re gonna try toe ultra bracelet, manipulate what we have indolent. And yes, So the first thing we&#x27;re gonna do is we&#x27;re gonna split the limited to two parts. So what&#x27;s that? Sorry, but something we&#x27;re gonna spoke with on the inside of the limiting to two parts. So the first part is gonna be sign theater divided by one of the second part is signed wanted, but by scientific data, Well, we can do is now We have a beta of the inside of the side and we want a data that did not deter. So what we do is we multiply by, say it open data. So remember this is valid because fairly divided by fate has one same thing. We have a tooth ada in this sign right here in the development later. So what we can do is become multiply by two theater divided by two. So finally so now what we get We moved that date all the inside So we have side to side of the divided by data We have a food and on the outside, you hear we have a two divided by sine to fade and we have Ah, what divided by two theater on the outside And the first thing we notice is the weekend cat So the cedars So all this manipulation bodily Let this to say right, But we&#x27;re not done because the question tells us to determine that living So what we&#x27;re gonna put on the inside right now Inside the limit is one divided by two times scientifically divided by data times to theater divided by sine too. We&#x27;re gonna take the limit estate or goes to zero. So but 1/2 is a constant we can&#x27;t really taking the limit of it will only result in 1/2 Science fate over sailors once the living a better one two things are divided by sine to date of the second this for so we have 1/2 times one times one. This is just 1/2.

Matt Just · Answer

Okay. See, we have feelinit Stada goes to zero of signed data. No contingent, Two things. And again, we just want to try to write this in terms of a scientist is over these so well, first of all, note that Ah, see, Couldn&#x27;t soco tandon data is co sign over sign. Okay, so we can write this in this kind of special way. So take the Hlynur because to zero I&#x27;m going to put a seita in a tooth data here and a co sign to theatre and wanted to do that because then again, But Scient, Ada, and then divide by a data set of that data cancels of that data and then the tooth ada over scientific data, which is coming from here. Guys, I haven&#x27;t changed anything, but this is going to one. This&#x27;s going to one. This is goingto one, and this is going to one half, so the limit is one

Issa Dababneh · Answer

All right, so in this western were asked the determined in the middle state it was zero off. Cynthia. Tough times. Contention to Veda. And we need to use the relationship. That&#x27;s right here. We need to use this these limits to determine the limit of the silently the tub school attention to All right. So the first thing we&#x27;re gonna do is we&#x27;re gonna try to algebraic Lehman appealing. It will have on the inside. So, uh, first thing we can do is we could sort of split thes. So who assigned to data time sign say that. And since we have signed to theater the denominator, that&#x27;s just one divided by sine of two things. All right, cause I&#x27;m too theater where you don&#x27;t really do anything to it. But now we have sign off data divided by one. We want to somehow change this into science data divided by So what we can do is because multiplied by seed in the numerator and state and denominator, remember this vicious valid Because they did anybody by fate, it&#x27;s just one. And now we go to the one divided by side of to theatre. We want to theatre in the numerator, right? Because we have to later, inside the side, we want in the numerator two feet. So what we can do is we can multiply by two feet are divided by two, because again, this is just one. All right, so no, what happened? So we have signed. So here we can we can take data into the insights. And I We have signed data divided by data. But remember, we&#x27;re gonna have a seat on the outside and again here. We can take the tooth aid on the inside, so we have to think of the numerator, but we&#x27;re gonna be left with one divided by two. And then now this is the algebraic manipulation. What we&#x27;re gonna do right now is we&#x27;re gonna put that back into the limit, try to Steve descriptive symptoms. So the limit as expressed is Europe co sign to fade. I will plug ins, Europe birthday. Do you gonna get co side of zero? Which is what? And then we know science data divide my theater is what, and this over here is also what? So what? We&#x27;re left with this data times one over two terms. Well, we have a state in the numerator. And if they did, they did a meter. So these cancel out were left with 1/2. So that&#x27;s our limit as exclusive is here.

Frank Lin · Answer

it&#x27;s clear. So when you read here, so we have the limit. State approaching zero. Sign Peeta over Data plus 10 gin. This is equal to when we convert it in terms of sign and co sign Tine over. Data was sign over. Go sign. We multiply both top and bottom by one over data and this gives us the limit. Must date approaches. Zero for a sign of beta. Over. Beta over one plus. Sign the data over. Data Arms one over. Co sign data. This equals the limit. State approaches zero. Signed data over Rita. Well, over the limit. Estate approaches zero. The one plus the limit. State approaches. Ciro. A sign Data over data earns the limit. State approaches zero. Everyone over co sign. You know, we get one over one plus one. Tell us. Excuse me. Multiplied by one over one. And this gives us one, huh?

Problem

Find the limit. $ \displaystyle \lim_{x\to 0} \…

Problem 45 Hard Difficulty

Find the limit.
$ \displaystyle \lim_{\theta \to 0} \frac {\sin \theta}{\theta + \tan \theta} $

Answer

$\frac{1}{2}$

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James Stewart

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