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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.
$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}-2 x}{x-\sin x}$

Carlos Pinilla · Accepted 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

Amrita Bhasin · Answer

we know this function is in the form zero over zero. Therefore, we know we can use the hospital&#x27;s rule. We know this is equivalent to zero over zero. What this means is that we apply the hospital&#x27;s rule second time. The two is now a constant. Therefore it just completely cancels. This is equivalent to zero over zero We apply The happy tells me we&#x27;ve gone. See that now it&#x27;s e to the X plus feet of the negative acts over co sign acts Plugin We end up with two over one. This gives us an actual answer of Jim.

Carlos Pinilla · Answer

so we want to look at the limit. 60 channa from the left is here. So yeah. Zero here. Here. Always staying positive. Approaching from the side. Oh, uh, sign of Lex Natural love even. Thanks. Uh, he&#x27;s not here. You need to take this. Lean it because well, not roll over. Move ex Somebody find for explosives So you don&#x27;t know Extends to approach from the side because from the outside is not defined. Um, so, uh, recalling hosts natural origin. Six goes this year. Oh, those mornings, impunity. Also, this expression will go for X equals zero sign of 00 but zero times. Uh, security also, you don&#x27;t know. What is that? So way. Need to make polite these to wait until more Something that we can novel. You hear? So, uh, well, we can, right? Signed us divided. Invite one over close. You can was divided by 17. So sign of Thank you. One is equal to one over one over several weeks. You look sexy. Photo one. Order one over sign of pecs so we can see these expression matured over the Multex time signing picks us matter loving mop X divided by Cosi tanto Becks. So, uh, ladies will sign. We&#x27;ll go. Whatever sign will goto if unity So this will be over the farm. Infinity or beauty. Uh, so that these expression he&#x27;s won Can I play Lopatto So apply Love it all. So that means Well, differentiate these in the friendship the bottom. So they&#x27;re leaving these people to really mean it to the same place sakes. Close, though, See, uh waas Oh, that&#x27;s good. Not alone. He&#x27;s one over X. There you go. Go Second is my nurse Court tangent Times co secret. So, uh, I&#x27;m going What is this expression? This is, uh Well, this is no other thing that will sign over sign Good times in that No sign of banks who external fix on, uh course he got these one. We&#x27;re X whatever. Sign of X. And so, uh, sort of these questions. He&#x27;s equal to one or ex. Divided by co signer Becks. Thems divided by sign XY squared so we can multiply his dad. I&#x27;m not right. Simplify diffraction. So So we get you Science Square goes on top. We&#x27;re gonna have Sunday school over big stems. And so well, at this point, can we just begin are so sex goes to Syria. All this gold zero I&#x27;m miserable toe one times zero. So they will go to Syria over zero. So at this point, we can&#x27;t apply loved account again. Uh, so can she. Duh that. So the idea of people sign score of X is gonna be two times sign affects. Really terrible. I&#x27;m now times irritable side affects the school sign breaks. All right, So? Well, people, to death on the little of this will be a problem. So different shape. First the storm, That wolf. Thanks. Just one. So be one times co sign No. Plus, keep that one constant x times 1,000,000,000 people. This second term in the birds. Verity. Both cool sign cheese, huh? Negative sign. So you have these expression? Uh, well, we&#x27;re Don&#x27;t forget that we&#x27;re looking at a linear sex goes Sierra Leone, right? Thio. We had to play with little twice, but he was not so, but

Mengsha Yao · Answer

Okay. Find the limit you stuck up. Thus ruler a proper it. If there&#x27;s a moral dimension, my third, consider using it. If locked US rule doesn&#x27;t apply. Explain why. Okay, Here we need to find the limit off dysfunction. And first we can plug in zero to dysfunction that is e to the zero, which is one minus one and then minus zero. So the top zero and divided by plucky zero to hear that hiss. Dear square, which is general. So this is zero divided by zero. I am determined form so we can use the locked house rules to read it as the limit as X goes to zero that curative off the top, which is e to the X minus the curative off a number one that hiss Cyril. So minus zero and then minus the curative off acts that hiss one and then divided by the d word half off. Ask x square. That is too X. Okay, here we get a new function and you can Stu plugging zero that is e to the zero, which is one and minus one. He wanted by two times zero. That is zero. So you get another Neil Function zero divided by zero, which is also and determine the phone so we can use the lock house rule again and that his limit as axe goes to zero the curative off the top. That&#x27;s just e to the X, then divided by the curative off the bottom, which is too so here for this functioning, plucking zero that hiss eat with zero it&#x27;s one and divided by two, so the answer is a half.

Bryan Lynn · Answer

already. So here we have the eliminates X approaches zero of either the X minus one minus X over X squared. So we will start by evaluating the limit. And it&#x27;s gonna be eat of the power of zero minus one minus zero over zero squared Eso. This, of course, is equal to one minus one minus 0/0, which is gonna be 0/0. So we can file a Beatles rule. And I know that this original limit will be equal to the limit as X approaches zero of the derivative of this top limit. So the derivative of E to the X is either the X, the derivative of minus 10 and the derivative of minus X is gonna be a minus one and then take that over the derivative of X squared and that will be two X. So now we&#x27;re going to evaluate this limit, and in that case, we get E to the zero minus one over two times zero says could be one minus 1/0, and again, we have 0/0 so we can take low petals. Rule a second time. Okay, It&#x27;s when we do that, we get that. This is gonna be equal to the limit. As X approaches zero the derivative of either the exes. Just eat of the ex. The derivative of the one is gonna be zero So far numerator We have just e the X and then when we take the derivative of two X, that&#x27;s gonna be equal to two. So now we can evaluate the limit again, we get either the power of zero over to just gonna be equal to 1/2 and that is the solution or limit will be.

Carlos Pinilla · Answer

so I want to look at believe it. Six goes to miners and ability off excess squirt TVX Uh, so you just Oh, him again. You know, the squared. Well, those two immunity, But you didn&#x27;t Minus infinity goes is here. So here you&#x27;re You have, ah, two functions. Oh, well, our fortune became reflection growing. So what is? Okay, Zero. So you have to money pull it this expression to be reading to a form. But he&#x27;s What should a bill suitable toe with But the the living ah idea should be that the exponential she, uh, should win. So this one is what should lead the behavior. That let&#x27;s make that idea more precise. So you have this. 03 ex people toe. Go on over to the minus X. Right. So what we can write? These is a Georgian X squared, divided by might have sex. So, uh, well, if you look at the dealing it now in this far wall, this is getting change all that. It&#x27;s also about to Central Square Anything. I&#x27;m not here minus minus. Infinity is gonna goto. Faith is about give the form you can be or in treating, huh? So we have a well, you know, farm figure completed so we can apply Lobito to this expression. Oh, so we dad, either fine shape his fortune or different shaving that function. So if you do that older, it will. Because quarries works under that&#x27;s bullshit to mindless sex. Sees you didn&#x27;t mind sick. We have a factor of miners. One do the channel that&#x27;s mine was one because they&#x27;re sounding their function there. 96 It&#x27;s a little What I have is you. We&#x27;re going at this point. Well, you have, uh, minus very being full time, baby. Well minus infinity to the minor sex goes to she gets so began. Not something off the farm. If you did. You ever see a d so will begin a right pickle again because we are. You know, the only lead off the front video are leaving. So you always always make sure that the I&#x27;ll clean it up before your meeting or zero over. So we&#x27;re in good shape till my lovely bones old definition that or that not leave it will equal toe the limit. Well, the argument here of the loot where X is going to 1000 change explosive one&#x27;s beauty. A great book to eggs, which is to no minus leaving 1 60 miles. You didn&#x27;t mind sex boating, and you get a factor of miners want due to ditch rule. So he&#x27;s gonna be That&#x27;s people till two. We find the minuses six, which is, people tone Well, six would want both. One is impunity. Well, all these girls school minus infinity minus ones in these woods everything. So you go slow if you issue a bottom. Well, that too. Stay still. Divide. You&#x27;re buying 18? I don&#x27;t know issue. So this limit we&#x27;ll be able to hear, as you can see here, the the exponential Let me be here because, uh, exponential moves gross. Plus, they&#x27;re not mention blues fluster Who&#x27;s faster than many going? It&#x27;s important book. What will you normal guest Exponential. The experiential wings In terms of really well, who was faster

Problem

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

Problem 20 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.
$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}-2 x}{x-\sin x}$

Answer

$\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}}{\cos x}=2$

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$\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}}{\cos x}=2$

Biocalculus Calculus for the Life Sciences

Grace H.

Heather Z.

Kayleah T.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Heather Z.

Kayleah T.

Michael J.

James Stewart

Biocalculus Calculus for the Life Sciences