find-the-limit-use-lhospitals-rule-where-appropriate-if-there-is-a-more-elementary-method-consider-6

Question

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x	o (\pi /2)^+} \frac{\cos x}{1 - \sin x} $

Mengsha Yao · Accepted Answer

The question is, find the following limit. Use the laptop through, or maybe a more elementary method. So, first, this is a one sided limit. And for one side limit, we can also use the lockdown through as long as this one satisfy the condition of loved US rule and the condition conditions of the laptops rule is that first, this function the top and about him about differentiable. And it is obvious because the both from the both functions elementary functions. So the top and the bottom a differentiable and second one is that the curative of the denominator is not zero. So first, let&#x27;s let&#x27;s find the directive of the of this one minus sine X, and we know that the directive of this is minus co sign X. And if you plug in pi over two here, that is minus pi minus coastline pi over two, which is zero. So in this way, this is sterile. So maybe in this way we cannot use the laptops rule. But, um, we can first write it as limit as X goes to hi or two, plus the curative of the top, which is minus Sigh X and divided by the curative of the bottom which is minus coz I x so minus minus has been canceled. So this is the limit X goes too high or too Plus sigh x divided by coz I X which is the limit as X goes to pie or two plus Condron hacks. So here the function becomes Tangent X And then in this in this way you can plug in pi over two and we know that the graph of, uh of tangent axis like this is like this Mhm, sure so And this side is pie or two and this function is periodic So the the next one will be like this. So here X goes to pi over two from the right that this X goes to Piau to like this. So as X goes to pi over two from the left from the right, the function goes to negative infinity. So we know that this limit is negative. Infinity

Bryan Lynn · Answer

All right, so here we have the limit as X approaches pi over to from the positive side. Uh, co sign of X over one minus scientifics will start by evaluating the limit at pi over to. So if we do that and we presume that this is equal to the CO sign of pi Over to over one minus sign of pi over too. We end up with 0/1 minus one, which is 0/0 swill. Apply low Beatles rule here. Include that this is gonna be equal to the limit as X approaches pi over two of the derivative of co sign, which is gonna be negative Sign of X over the derivative of one minus sign of X, which will be so the one will drop out and we&#x27;ll get negative co sign of X. And, of course, then these two natives are going into simply cancel each other out and sign over. Co sign is, of course, just tangent. Right, So this is really equal to the limit as X approaches pi over two of the tangent of X OK, and so we can evaluate this and now Tangent of pi over to is undefined. Remember, we&#x27;re coming from the deposited end. And so if we wanted to remember what a graph of tangent looks like high over too. It&#x27;s gonna be something like that, right? So we are approaching negative infinity here. Therefore, the limit as X approaches however, to of tan X should be equal to negative infinity.

Carson Merrill · Answer

So once we directly substitute X equals zero. What we end up getting is 0/1 minus one. And that&#x27;s clearly just 0/0, which is an intermediate form. So we are allowed to use local towels role when we take the derivative of the numerator, What we end up getting as a result is um a derivative X squared is going to be two X. And then the derivative of the denominator. Since we have one minus cosine X, we end up getting just Synnex. Now we can plug in zero. Although this is just going to give us 0/0. Again, another indeterminant form. But the beauty of locales role is that we can continue this. So now um using little zero, we take the derivative of this, we get to we take the derivative of the bottom, we get cosign X. Then based on this we can plug in zero. And that&#x27;s going to end up giving us two divided by co sign of Zeros one. So our final answer is going to be too

Bryan Lynn · Answer

Okay, so here we have the limit as X approaches infinity of x times sign of Pyrex. And so we&#x27;re gonna rewrite this so that we can use low Beatles rules. And so this is going to be essentially the same thinking as the limit as X approaches infinity of high times. Sign of pi over Rex over. Pi over Rex, Right? And so what happens is these two pies would cancel out, and so this is the same thing. But now when we evaluate at infinity, we get pi times thes sign of pi over infinity over pi over infinity. It&#x27;s That&#x27;s gonna be, uh, anything over infinity. We treat like a zero. Okay, so you have pi times sign a zero and sign of 00 And then this pilot a penny become zero and that that is in the form of low p tall. And so the crack into terminate form s O. We can go ahead and use low Beatles rule. And so that means that we could take the derivative of the top and the bottom. So this is the same as the limit as X approaches infinity of Yes, we take the derivative of pi times Sign of hire Extra pies Just a constant suits Pi times the derivative of science Have Rex that&#x27;s gonna become co sign. Uh, hi over x times. Negative pie over X squared. Okay. And then when we take the derivative of pi over X here, we&#x27;re gonna end up with negative pi over x squared when that we have a common factor. So we we can cancel out this native pi over X squared, negative, high over X squared and just concern ourselves with the limit as X approaches Infinity of pie, a Times co sign pi over X. Okay, So when we evaluate that high times CO sign of high over X Ah, But at infinity so high over infinity, that&#x27;s gonna be the same as pie A Times Co sign zero and co sign a zero equal one. So this is just equal to pi and then that is our solution

Carlos Pinilla · Answer

So we have the myriad six. So zero off the function e the board eggs minus the minus x. Oh, lives to eggs over X minus. Sign off books. Well, so what happens if he just go ahead on play here? The place for X equals zero. So here, we&#x27;re gonna get They&#x27;re gonna get one of these here. It&#x27;s just one issues one. So one minus one and they will to them. Zero is just you. Why, I might as one over minus zero. Over. Well, zero on the design of zero. So zero. So you&#x27;re gonna get something over the farm or what does one is, you know. So you&#x27;re on, Your Highness, your zero. So your orders. You, um So here we&#x27;re We&#x27;re in good shape, Toe. I love you, Joe. You&#x27;re here. I love you, Joe. So what they don&#x27;t tell us is that these need will be ableto the limit. Sex goes through. Same, please. Zero. But you different shit. They if I weren&#x27;t She So there is about eating the exits. Just you did the X veritable getting mine six will be You didn&#x27;t mind in six, but, uh, here we have to be careful that there is on your function there. So by the channel, you get out mind as one coming out off the function minus six. So these mind, this is what walking playing might see DDX on the Red Bull. Twix, She&#x27;s still so my still I&#x27;m game home bottom here with that gettable exit. Just one. They&#x27;re a team of scientists. Go sign. So one minus. Oh, son hooks. So all these simplified will be through the X. Okay. Minus minus plus minus six minus two. So well again. Well, you just blogging. X equals zero. We&#x27;re going out. He is here about Swann. Plus you didn&#x27;t mind. Syrup is also one. It was one minor. Still, that is your here but one minus because I&#x27;m of you is one. So again the well we perform so well. We can&#x27;t go ahead on apply. You know there is no for being you. How many times you gonna buy local? So lactose is that these little people toe limit off ex going to the same place but ready one top. Where you my dear bottom. So they&#x27;re atrophied. Three exits. Just x Remember here that these mind knows he wanted playing that. So that&#x27;s how blasts you gonna wait? Six plus, you didn&#x27;t like six. Then again from the channel rule from the minor sex there come south, another miners wander and then they would look tow this year. So my seal, then the girls wanted Cyril. That live off minus co sign is, uh, minus sign so minus sign, we&#x27;ll fix so well. Here you have. You know, whole X here is going to show you your place here expletives you these side of zero zero. He&#x27;s the one in that. I&#x27;m saying there because he was one minus. You did the minus here, which is one so again you would have something of the farm zero over zero. Well, you can go ahead and play love you go one time, so keep on. They&#x27;re due in our game way we know to be friendship could be excessive if you&#x27;re pretty, ext t x minus. You didn&#x27;t wait? A six will be minus minus X multiplied by your minus one. Coming from the minor. Six. I&#x27;ve been here, but well, that&#x27;s this is a minus on the minus of that&#x27;s plus sign sign is a sign. So these accounts for there Developed they So it&#x27;s the bottom? No. So at this point, it seems a very you can get something because we&#x27;re dividing by something that does not go to zero, cause I know zero is one. So at this point, we come forget and focus on a zero. This one, my nose of miners who get a plus So one they did that minus plus what? It&#x27;s just gonna be you go to two over one. Just one. Just resist yet. So I used to thio because he gets to apply though waas wise through. That&#x27;s but we finally got a deal. Beaches step. You have to make sure that you&#x27;re, you know no, leave me. That is, uh, forward for which is likely to apply locator So big toe the farm zero or zero Or security or if you

Carson Merrill · Answer

Yeah, this problem, my hair, we&#x27;re going to be using what cows rule to find. Uh the limit of the function. Natural log of X. Get that uh times the tangent. So we&#x27;re going to do over the co tangent of X. High over to. So that way when we plug in one plus we end up getting a negative chew over pie. So in this case we don&#x27;t need to use locales role as long as we convert it to this form right here, once we plug in the one here we&#x27;ll get pi over two. Um And then this will be uh Approaches one from the right, we&#x27;ll get Natural Log of one. So this is through taking the derivative here. Actually we do have to take the derivative because we get 0/0. So that&#x27;s one over X. And then this is going to be negative coast second squared uh exp I over to you Time high over two. Then when we plug in one we would get two divided by a negative high

Mengsha Yao · Answer

with this problem. We see that if we just plug in um pi over two directly, we would end up getting the indeterminant form. So we see that lope tells rule is applicable. So we&#x27;re going to take the derivative of the numerator. So the derivative of cosine X is going to be a negative Synnex and then the derivative of 1- Synnex is going to be negative cosine x. So based on this, we have negative Synnex over negative cosine X. That just equals tangent of X. Now that we have the tangent of X, we can evaluate this as X goes to pi over two From the right. So let&#x27;s find where x equals two pi over two. We see it right here. And based on this as it&#x27;s approaching from the right, it&#x27;s clear that we are approaching the value of um negative infinity because this is the graph of tangent of X. So, right, Because this is the tangent of X graph. We see that at pi over two, the tangent of poverty does not exist. So based on that, we see that our answer would be a negative infinity. As the final answer

Problem

Find the limit. Use l'Hospital's Rule where appro…

Problem 13 Medium Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to (\pi /2)^+} \frac{\cos x}{1 - \sin x} $

Answer

This limit has the form $\frac{0}{0}, \quad \lim _{x \rightarrow(\pi / 2)+} \frac{\cos x}{1-\sin x} \stackrel{\underline{H}}{=} \lim _{x \rightarrow(\pi / 2)+} \frac{-\sin x}{-\cos x}=\lim _{x \rightarrow(\pi / 2)+} \tan x=-\infty$

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This limit has the form $\frac{0}{0}, \quad \lim _{x \rightarrow(\pi / 2)+} \frac{\cos x}{1-\sin x} \stackrel{\underline{H}}{=} \lim _{x \rightarrow(\pi / 2)+} \frac{-\sin x}{-\cos x}=\lim _{x \rightarrow(\pi / 2)+} \tan x=-\infty$

Calculus

Grace H.

Catherine R.

Kayleah T.

Michael J.

Integration Review - Intro

Definite vs Indefinite

James StewartCalculus

Grace H.

Catherine R.

Kayleah T.

Michael J.

James Stewart

Calculus