Problem

What happens if you try to use I'Hospital's Rule …

02:58

Problem 46 Hard Difficulty

Rank the functions in order of how quickly they approach 0 as $x \rightarrow \infty$
$y=e^{-x}, \quad y=x e^{-x}, \quad y=e^{-x^{2}}, \quad y=x e^{-x^{2}}$

Answer

$y=e^{-x^{2}}, y=x e^{-x^{2}}, y=e^{-x}, y=x e^{-x}$

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Calculus 1 / AB

Biocalculus Calculus for the Life Sciences

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

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Video Transcript

All right. So you have the functions he did in my six function eggs e didn't find six. This little Jim, You didn't mind Sex? A squirt in the function X. I'm seeing that in weight of six square. So you want to arrived? Them us Hope first, did he go to zoo us? Exclusivity. Well, so, uh, in this single a statement off these question the Subaru go to see you so well. But this function is clear that it goes through serious explosives ready before that function as well, Because, well, you just take the limit. These excellent security, it's clearly is you. But now what about that one? Does that don't really go to zero. So we have to delimit sex goes to your vanity off ex either to minus picks, so well, move it. VD them zeroes off. Why is that? So what you can do is, uh, with me to love little form. Uh, so I'm just rewriting them x over e to the minus six, so well, here we have logical for him because says go stupidity. This goes. So the form your affinity, huh? Means the sign there. If I send it down huh? Listen, minus sign. But so it's gonna be with the foreign community or a tear in Italy. Explosive very gets exhausting reading. So all differentiate. So you get one. Who are your three ex? Which orderly goes through? Most issue was one over baby before Weibring TV. So what's this? You is the media people on the similar story with its function. So what is the limit? Sex goes to infinity Fix. I'm seated in one of six squirt. Uh, so again, Same thing, right? Him, us Excuse Today. Extra square. So our sex go cpd form d for you, baby, Is, uh we go there s o you apply love. It'll to that. So that we need will be ableto the limit. Six goes through the same place. Katie on the theoretical, These one over. There you go. That exponential he's got of us. My legs. Oh, me is there. There is no mine, is there? So Well, uh, it's so deformed. One of our infinity, So same thing goes, is you a little bit or good in all of these girls to go to sea. Now we try and goes faster. Uh, so these ones should be the one burn faster? No. Um, the next function it should be easy onto. See how that should be true. We compute the limit. Will you do the questions? And so the If you put there be your function on top, then the so will were The slower is low is low, the more slowly became function duck dinner. There's a limit. Six goes to infinity. She go to Okay, so if you put there no wonder the decreases slower is the meat should be you think exclusivity and watch as you can see single people Computation there was equal. This name is equal to the limit off X Which six ghost radio. The nearest off axis. Um so you have that. And then the next function should be should be these one. This one should be their third in terms of which one Nic Cage buster. By the same reason, If we do the limit off the question, doesn't it two functions, But in the one that decrees decreases is lower on top. Then we should get mm. I should get in. So while these can be reading like that eso this is it exactly the same Limit Sex ghosting pretty. Oh, uh, well, we can send that on you Did it. Excess squirt line us looks We're X so well, let me tell you everything us well, we can we can differentiate from, so these will be open for me. Well, exit squared where No six leave it. Sex goes through your day off. This is equal to punitive because this name it is off the farm. Six roast in Haiti off X squared. Them's one minus more bricks. Right? So? Well, this leaving would be ableto the purple, the limits giving that that they exist. They now for these limits for the limited, right? Most part, the limit is one who are other ex. So this really is one. So it's gonna be okay, So? Well, uh, if we see that these way, we can see that. Yeah. Hello, petal form. Because these terms, we'll go to you, Katie. So we'll be friends. She that Then why Lopatto? We get and supported Limited six goes through 80 off the top of these would be Well, there go the top, Which is he's gonna be to explain this one. See you there. Excess HQ word. Mindless sex. This term comes from the general on the bottom. You got one. So the dream it will indeed go to video eagles. We don't. This will be infinity times. Infinity. Uh, so, yes, these function, uh, dysfunction, which is the one that we wrote? What? It was this So here, dysfunction. We wrote that function on duck that function without and it gave us You need to accept. That means that this function decreases more slowly. Three decreases more lonely too. So you have three What was being done to which is bigger, that one. And now for these last case, we can do eggs. You can't do the same argument this before it could be more the most. The lowers slowly, the more is only decreasing function on top. So well, what we should display with the addition of you, you feed the six boasting feeling like in your prediction is correct. This should happen, Ana. Well, in dust because that counsels with that. So this is what you get that there are getting functions is dysfunction. Is there foster First Sergeant this second function in terms of our twenties? Because this faster is this one. That third function is needed when the sex and then and the reason is that having a power on day on the export in the open exponential you see, you know, better decree decrease rate than just well, clearly they x here is exclusivity messes things up in terms of what is there. What was the case? Faster, I think because the only difference between this story is that this one has an extra X so that makes you dysfunction decrease in faster. But even if thes one has on eggs, I think st and this one does not. This one does decrease faster. Do The quiz is faster than three. So they The main thing is that the exponents through the square is what makes you the crazy buster.

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