evaluate-each-limit-if-it-exists-use-l-hospitals-rule-if-appropriate-lim-_z-rightarrow-infty-z-e-z

Question

Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).
$$\lim _{z \rightarrow \infty} z e^{-z}$$

Amrita Bhasin · Accepted Answer

first things first. Let&#x27;s rewrite this without the negative exponents. It&#x27;s easier to read now. We know that we have number and denominator both go towards infinity as X goes towards infinity. Therefore, we can apply the hospital&#x27;s rule. Therefore, take the derivative of top and bottom three X squared over two acts each. The X word you know you can cross off one X Therefore, we have three acts on the top over to eat of the X squared. Now we capture a pile hospital&#x27;s rule again In order to actually get an answer, we end up with three over infinity. Remember anything divide by infinity is zero.

Bryan Lynn · Answer

do we need to solve the limit? As X approaches infinity of X cubed times e to the power of native X squared. And so each of the power of negative X squared is really one divided by Ito power of X squared. So when we rewrite it in that form, so it would be the limit as X approaches infinity of execute over e to the power of X squared, Then we can he violate this limit. So let&#x27;s evaluate at infinity and we get infinity cubed over e to the power of infinity squared, which is, of course, infinity over infinity. So this means we can use Blue Beetle&#x27;s rule here and take the derivative. So let&#x27;s take the derivative, uh, both the numerator and denominator. So the derivative of the numerator being three x squared and the derivative of the dominator being two x times eat the power of X squared. You know, we actually get a cancellation, right? We can get rid of one of these exes us. This is going to be the same. Oust the limit as X approaches infinity of three x over two times e to the power of X squared. So we will evaluate again and we&#x27;ll get three times infinity over two times E to the power of infinity squared, which is infinity over infinity again. A solely used Loki tells Rule one more time or who knows how many more times, but at least one more time. And so that will give us the limit as exit, purchase and Fanny. Now you want the derivative of three X, which is three and then the derivative of two times each, the power of ex squares that&#x27;s gonna be It&#x27;s gonna be two times, two x times each part of X squared, which will be four times E to the power of X squared. Okay, so now we will evaluate the limit again, and in this case, we&#x27;re gonna get three over four times E to the power of X squared. But not an X ray. We&#x27;re evaluating it. So it&#x27;s gonna be infinity squared, which is equivalent to three over infinity, uh, which is gonna be equal to zero. So the limit is equal to

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

Carson Merrill · Answer

the first problem, we&#x27;re going to be taking the limit as X goes to infinity of the function. Um So in order to do this one, we&#x27;re gonna want to take the natural log of both sides. And when we do that we can simplify it to be the limit as X approaches infinity of the natural log effect over E to the X. The reason why we do this is because now it makes it much easier to evaluate the limit. When we plug in X going to infinity, we get infinity over infinity. That&#x27;s an indeterminate form though. So we&#x27;re going to need to use local tells rule the derivative of natural log of X is one over X. And the derivative of E to the X is just to the X. Now, when we plug in infinity we will get here. One over infinity is zero. So it&#x27;s got zero and then each of the zero is one, Even the Zeros one. So we get one right here. But zero is not our answer. In fact, the natural log of Y equals zero is our answer, because we took the natural log of both sides. Now, when we solve for why we see that Y equals E. To the zero, which equals one, so one is going to be our final answer.

Carlos Pinilla · Answer

So we have a limit. Essam, you goes through with a function? Yeah, to the bar. You You&#x27;re right. 10. Or you borrow three. So Well, if you go ahead and just plug you Yes, you go safe for all of you. Wait. 10. That&#x27;s gonna goto well where he goes. Exponential wolves? Definitely. So? Definitely. So you you was gonna go if you are. Time&#x27;s gonna goto. Verity. I&#x27;m Dana Borromeo also grows. They finish you okay? Indefinitely. So he&#x27;s gonna go to you lately, So we&#x27;re in good shape applying. Let me go. Well, you don&#x27;t, and so will we. You can go ahead and different ship dysfunction. One of the friendship Exponential. You get the same explanation, but, uh well, be careful there. Here you have. Ah, on our function will you compose him inside the expansion, which is? You did it out. So you get a fuck off. I want them additional special. So these would be this. But it is inevitable. You you over. So don&#x27;t eat it. You the either you were 10 time&#x27;s. One thing you overcome bonded with the friendship, the water All the power comes own. It comes on their plane when you lower before by one. So you get three times you squared. Well, this squared is 3,000,000 won. Um, So what rule tells you that she&#x27;s leaving Will be exactly the same. Was that lead chair or obviously terrified? There are the community in us. Yeah, you We can when you send you 10 on bottom times. Three years old. Three times. Stan, That&#x27;s Trudy. Comes, you squirt. Well, let me fix you. Still may be, uh so as you can see here, uh, yet plugging because I would have she goes there, baby, he&#x27;s gonna go to infinity in under three Times Square. We&#x27;ll also go to 30 times Your score&#x27;s gonna go 30 times infinity halting TV. So we&#x27;re also you love it. So Lovato framework, uh, so we can apply looking again. So when you differentiate this expression on top, you&#x27;re gonna get Yeah, you advance times wanted begun. Chiharu from the Times 30 on the power Here comes own. So it&#x27;s ready. Times two the four of you decreases by one. So So it seems you won so putting off their those one over. Well, then 60 three times 36 So you get um, one of their six hundreds, uh, e b b you dense or you. But at this point, you just brilliant. Then you&#x27;re gonna have the form here. Well, that&#x27;s gonna go to Katie you Folsom Treaty. So we need to play lobby till again. So this limit will be brutal. One word, six Science. You can put that order willing. It lead us. You those to infinity. Oh, you today, you over 10. Do you dance? Men were beautiful in shit piece. You get more than facto wanted. I&#x27;m there. Well, there are two of you. So these guns, or did I read? People Get up. I&#x27;m gonna forget their water off the bottom. They&#x27;re illegal fuel. Just, uh, what is one? Um, is gonna be one over 600 stance, then. So one of her six thousands, but, uh, So you go see if any titties over water help. Is there being on the bottom This girl boy? Either way, So you want to get beat? Well, you&#x27;re gonna be in a water hole. How many thousands longer. It&#x27;s it is final. No. What? You got to clear the air. Well, the morale of these long story should be, Huh? Uh, in the exponential Rose Foster, Go around. Not a play on. I don&#x27;t want a ho peak. So you have your reporting on you Bottom and exponential Will&#x27;s work faster. So these will our impunity this world were ween the bottle. Which function goes faster? Does getting you that, uh, the lead issue goes to Haiti? Those fools, You can&#x27;t have that expression.

Amrita Bhasin · Answer

the problem. This Find the limit you slip. It has rolled wire shock. Verity. If there is a more modern tree, my third can stay there. You than you. If you will pass through, it doesn&#x27;t apply. Explain why the limit X goes to infinity. X Q Hams Each objective at Squire this is Cho Choo limit. X goes to Infinity ax Q over you to exploit what that&#x27;s goes to infinity, Not you. Start X cube and into exploiter or to infinity so we can use the casserole. This&#x27;s a cultural lemons X goes to infinity. Three ex squire hams over into X Square has two acts. This is the continent Levitt Ex calls to infinity three acts over to house to exploit, then use lapis casserole beans. This&#x27;LL ease come true Limit. I discussed Infinity three over to Nancy to X squared times through packs. So what? X goes to infinity? The denominator goes to infinity. The answer is zero

Problem

Evaluate each limit (if it exists). Use $L$ Hospi…

Problem 18 Easy Difficulty

Evaluate each limit (if it exists). Use $L$ Hospital's rule (if appropriate).
$$\lim _{z \rightarrow \infty} z e^{-z}$$

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