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Question 21ptsLet f(x) = [2x2 + x . Find a formula for the Riemann sum 2 f(c)ak obtained by dividing the interval [0 , 2] into n equal subintervals and using the ri...

Question

Question 21ptsLet f(x) = [2x2 + x . Find a formula for the Riemann sum 2 f(c)ak obtained by dividing the interval [0 , 2] into n equal subintervals and using the right-hand endpoint for each Ck(2u + 2)(7n + 8)(Zn+ U(UZn + 8)(2n + Tn + 6)(2u + U(Tn + 6)

Question 2 1pts Let f(x) = [2x2 + x . Find a formula for the Riemann sum 2 f(c)ak obtained by dividing the interval [0 , 2] into n equal subintervals and using the right-hand endpoint for each Ck (2u + 2)(7n + 8) (Zn+ U(UZn + 8) (2n + Tn + 6) (2u + U(Tn + 6)



Answers

Find a formula for the Riemann sum obtained by dividing the interval $[a, b]$ into $n$ equal subintervals and using the right-hand endpoint for each $c_{2} .$ Then take a limit of these sums as $n \rightarrow \infty$ to calculate the area under the curve over $[a, b]$ $f(x)=x+x^{2}$ over the interval [0,1]

Ah, good day, ladies and gentlemen. Today we're looking at problem number 48 in section 5.2. And it is it is asking us this function path of axe equal three X plus two that squared over the interval. I will. 01 We're supposed to find the end, um, reminds some, find a formula for it and then use that formula to calculate the respective limit. Okay, so I'm in the previous problems. Ah, you can check the previous problem on this number 47 a few of those prior to that one. I actually explained, um, a lot more about the the remind sums of stuff that I'm going to do this time. This time I'm gonna just get more to the, uh, the guts that got to the problem, if you will. So I'm not gonna explain the whole idea of the remind. Reminds some this time. So if you don't understand what's going on here, please checked some of the previous slides because some of the previous problem I did a lot more of what three months sums are. But this time it's gonna get I should uh um I'm just gonna cut to the chase. So, um, so we're given and okay. And what we're what we're after is a formula for s n. Okay, so we're after we want to calculate, um s and, um, which is equal to and this and this. Please, just look at the text book. Ah, it's actually in there, but the formula, uh, for the s and, um, looks Well, I'm sorry. Out a little bit. The s n actually looks like this. Um, this is the form they give you in the textbook. Uh, it goes from a equals one, Thio, Um, two in here. Um, um um, and what it looks like is, um f death of X k times. Dulce. Ah, uh, that's k okay. And, um, we want to find a formula for this guy. We're using the right hand and points, and our Delta X K is equal to, um, one over an air. So it's equal to one over and groups it's equal to it is equal to one over end. Okay, so, um, if we're given and just picking a x k, um, this, uh, to get the formula for it, we just plug in. Um, we just plug in. I want the formula for this guy year. So the ex k um, hers. Sorry, the ex k, um, taking k over and into this formula, we're going to get three. Hey, um, over and huh? Plus, um uh, two times k squared, divided by, of course and squared. And again, I'm just replacing I ke here on the x k here. Uh, yeah, boy, the ex k Maybe I don't want to. So the x k here I'm taking is, um hey, over n Okay, So that's what that's what I'm That's what I'm putting into this form of bigger. So it is this, um, multiplied then by the delta, which is one over n So it's Delta hopes I'm so it smelt multiplied then by one over n and this year is my This, um, is my s n. So this is s. And it's equal to that. Uh, And now I can break this, and I'm gonna break it into two separate sums. So I break it like this, um, into my two separate sums, just breaking it over the some expanding the this the, uh So this would be a plus. Then the second some here. And of course, the 1st 1 is when I'm multiply through by tthe hee won over an I'm gonna get, uh I'm gonna get three k over and squared. Um, and the other one here, I'm going to get the two k squared over. I'm gonna get to K Squared over and you So again, I'm just separating this and dividing through by the the end here. So I'm just simplifying the stout here. Okay, So this is my formula for s and at least my first formula for SN. Now, what I want to do is I'm going to simplify. I'm going to simplify this further, and this is what I'm doing. The next slide here. I'm gonna take this, um, and using the, uh, the formulas in the book. I have, um, t I have the formula. Uh, okay. You see here. Hi. So, okay, goes from okay. Equals to one, two. And, um, this is, uh, this is equal to and and plus one divided by two. Yeah. Hey. And similarly, we can do the Saint Weaken. Do the same thing with, um, the end square here or the case square, So Okay, um, goes from one two, and, um Ah, and, uh, K squared k squared, uh, is equal to, um this is equal to groups. Uh, that's equal to and and plus one, um, two and plus one. All of this divided by six. Okay. And then taking, so ah, you'll notice that this is this is three k over end, so I'm just gonna multiply both of these sides here. Um, bye. I'm gonna multiply both of these sides now by three K over and so I get a hot not three over and not three k over and sorry, three over end. Um, maybe I should move that. Whoops. Come on. Okay, so the three over end here and the three over end on the other side pain. So no reply on. I do the same thing over here on DDE. He and I noticed right out the back that the end cancels O r. And get rid of the end here. And the lips the the end. Hearing, of course. Replaced up by 33 times. Get rid of this tree here. Um uh, Wait a minute. Oh! Ah! I must have made a mistake. Uh, right. That was not in. It was supposed to be and squared. I'm sorry. So you just like that didn't happen. That wasn't right. Um, and similarly on Ah, from Yeah, that's right. I'm sorry I took the wrong thing. Uh, I multiply both sides by the, um that tea over the two over and cube. That's right. Like, that's not gonna work out, right? If I don't do that s o I multiply on both sides by the to over and who's the third? Um, and, uh, putting that in here Ah, to over and to the third. Um, and I noticed right away. I can just drop that to us, too. Uh, get rid of O and get rid of the two and make that a one. So then this the denominator becomes three. And the numerator is just n plus one times two and one. Um, help me out a little bit here. I'm going to expand this to be Ah, this is gonna be too and squared. Um, plus three and plus one. No. Um, and now I want to some of both of these guys. So I'm I want to some of both of these guys. So when I do that and do some simplify. I get my s N year s and is equal to, um, three divided by two. Because I just, um, over here, I they factored that. And, um Oops. Okay, three divided by two. Um, plus sorry. Plus, uh, three divided by two n plus. And now I'm doing a lot of the same thing over here. Uh, which is that gives me a two over three, um, plus ah, to over. And, um, no, not to over. And, um sorry. One over. End two over and one over in one over and And then plus, um, 1/3 end squared 1/3 and squared. OK? And this is the formula for my sn. This is the formula that we're actually we're actually trying to get Okay, So this was the formula we're actually trying to get. And now the final step here after we get this far as we have to compute the limit as end goes to infinity. So we want to compute the limits. Um, as ah, and to infinity, um, of s in here. So off the s n. We want to compute that limit. And of course, that is equal to, uh, the limit of the left hand or the right hand side. And now you'll notice here that if we go through individually because we can break this into sums that, um, this guy, uh, goes to zero, Uh, this guy goes to zero, this guy goes to zero. The only one we have left is the three has and the 2/3. And so we get that. Our answer is equal. Chip clips is equal to 3/2 plus 2/3. Um, which, of course, is equal to nine over 9/6. Plus, um uh, 4/6, which is 13 over six. And that is our answer. So, um yeah, thank you very much, folks. Allow at work. That's our answer. So take care. And, um, good luck.

So this lesson in this lesson we use the room on some to find the area in there. That calf X plus is quiet from that one. So essentially, this is what we are finding. Okay, So the concept is that we have an interval from level 21 There's a cab over here. No, if you divide the the area, the overall area too. So money with heaven into so many paths having the same with we would be ableto estimate that the area's off that rectangles. Then we some all of them up. So the truck so we could have the rectangles like this. Okay, as it continues. So the length off each each with is one minus zero uba. And because you're dividing the whole thing so many times, all right, and that is equal to one over. So what it means is that if this is there the length from 0 to 1, each of them becomes one over n. So the right part, this one of the n Then if then if this is the next division, this becomes one of another. This becomes two times one wine that is to end. This becomes three times one over N That is terrible. And and on and on until we have an over. And that is a call to what? Yeah. Okay, So what? Each off this point, we will take a sample point seeking that is the most right. The very right off each interval. So as you can see, it goes from one of the end to the end. Three of the end and all that. So it goes from key, but And for each of the cave interval, that will take the first interval. Second interval fed fourth and all that and that is the right side off the interval. Okay, so the Roman, the Romans some they were months, um is some from 1 to 10 for F or seek a this point very off evaluated. Then times the length off the interval, I'll stand approaches in Trinity. A very huge end. Okay, so this is equal to the Yeah. Okay. So as we already know, the f off X waas equal to X plus x squared. So in place off ex you put key Seiki were there And that becomes key on n less que en or squared as we have here, then. Times one own n Because you've seen the length of the interval at that. So as an approaches infinity. Okay, lets go on. So now let's deal with Onley. Does some than we come to the limit. Okay, so car. So Cuba en last key over K squared over and squared than multiplying bad. Less identify are constants one about and it's a constant cuba in. Okay, then you also have one of and again just yeah, splitting them. Yeah, okay. Squared uba, um squared. Okay. From 1 to 10. All right, so we can take this. Came back there because the case a constant So we have vulnerable and squared then key from one to end. She then this comes back because a constant the constant is no affected by the the summation. Okay, so we can bring it back to be a multiplier. I already have one of the Ansel If one of the and squired Joyce, it's that is this us one uber and squared. If something like that, if a joints it becomes one of the n to the path three. Okay, and you still have the submission? Oh, he squared. Okay, so having this that is K goes to one and is the heaven one Last two last three last four last five on an own class in and this is and then the last one all over to that is the sum. Okay, if you you summed them and also happen this way. We're dealing with us some fast. Then this is heaven que codes to want em. 71 Last four last nine last 16. So yeah. So instead of 11. 12345 you have That's the squared off them. So we have 14 that is two times two. Yeah. Three times story that is 94 times 4. 16. 5 times five is 25. Known unknown. You have, um, the last one and squared. Okay. And that ISS then and last one to end last one. All of us. Six. That iss the submission off all of them So we can substitute this into that and this into that. Okay, so it means that the whole thing one of the and squired some mention k one to end. Okay, last one of our end to the path. Three summation, K one to end off K squared would be equal to Okay, let's do is yes. So that they would align on over and with one movie ends squared. Then in place of all this output in and last one all over to put over Sorry, the name place off the horse. And I put Onda last one to end last one. All of us six. Okay, so as you can see this to off the ends here and this one So we end up heaven one over and then one of the AIDS Council's this one. So let's write it appropriately. Okay, then. Yeah. And last one, two n plus one all over. So this end councils one of these. We have six and squared. Well, okay. Moment. This is what we have. I never Okay, okay. The next time we can do is to simplify. We have at this point, if you multiply the e f. And to that, you have to unscrew aired. Then you have end to this. You have n okay. And you have one multiply in that you have one multiply in one, all of us, six and squared. Okay, So what is? And last 1 to 10, then we have to end squared Last three and last one, all of our sixth and squared. Okay, so all of these is without the limit, if not taking the the limits. All right? Yeah. So, Heaven, this has an approaches in 20 K 1 for banned for a four C key. The X would be called to the limit as an approaches infinity off in class 1/2 n gig. Just splitting them. Okay. Okay. So one thing is that if the end is very huge, I didn't want to. It will know. Make any sense, Would know would not change it. Let me put it that way. For example, if e f 200,000 and you add 12 with divided by some other number me be no change so much as you had no add one to it. Okay, so if the number is very huge, the number is very. He added a small number to it. Almost make it off. No effect. So it means that at that point, the only significant numbers or only significant path off this will be the end and the two and other down. So means that this is a quote half because at this point would ignore the one doing this and will cross out the end. Then you only have So you would know this whole paths that you have half right? Then that's bad. Is that if n is very huge, all of this will, no matter. Adelaide to to end squared would make it as if we have no added than your thing less imaginary off three bilion than you ask. You are multiplying it by itself again. I didn't want to it or even three billion to it will know make any it will not change is significant. Least what means that there's a very high value would the only significant path will be to end to the power to Then that crosses. This will cross that. Ignoring this pod would have two on six. And that would be one on three. Okay, okay. So that ISS one on three probably talk about two and six. Then we can simplify it further as one on three. So here we have six. Two goes into sixth, three times 30 goes into 62 times and that is five on six. So the idea under the cab is 506 squired units. All right, Thanks for your time.

So this questions asking has to find a formula for the Riemann, some obtained by divine and rolled and equal stuff minerals, using the right hand and point reach ck and then taking a limit of these sums and approaches infinity in order to obtain the area underneath the curves. So first thing, any nose is exactly what is my formula for the room and sons and were given that that is equal to case able to wonder and of my function evaluated at ck times, Delta X The first thing and you find is exactly what my Delta X and CK is. So my Delta X is nearly the width of each rectangle, and I find that by subtracting my end points and dividing by my number seven rules, which in this case is and so it's just gonna equal one over end in my CK, which is the X value of where my high school me evaluated at. That's just going to be K times My Delta X. It's just going to K over end. So then plugging that and see my son whenever some from K equals one to end of my function evaluated at CK so ck wass k over end. So it's going to be three times K over and plus two times K over and squared times Delta X, which here is just one over end. So it orange to calculate what this sum is equal to. And when it just started distributing and figuring out exactly what we have inside here. Caves on end. This is just three K over end plus where I'm screwing a fraction. Both sides are Tom Bond. Get squared. Someone have to k squared over in suede. I was wonderin. Then I went to distribute that one over end. Soon I get three K over and squared plus two K squared or in cubes. Hey, it's Natalie and start using our summation rules. I can use my son role to separate this into different to different sons, and then also, I can use my constant multiple rule. So here, rebelling from Kate that kay So this and is actually just going to be a constant years. I can take my three over and squared and two over and cute and move them to the front of my sums. So when I do that, I have three over and squared from some of k equals wonder and of K plus two over and cute comes a sum of K equals one to end que ce qu'on And in the end, we have a formula for the sun. Okay, in the sum of K squared. So this first one is just going to be three over and squared times, Um, and shins and plus one, two, this was going to shoot over and cubes times and times n plus one comes to and plus one all over six And then I just do some last simplifications. So here in animal, cancel out. So I get three over and plus one over to N and cancelled out here suddenly and squared to cancel out so much with n plus one Tim and plus one over three in suede. So last part of that question is asking us to find limits and approaches infinity of our son. So what have Lim sent Purchase and Fendi of here? Do you have two different things? Some so I can buy limit laws. Just take the limit of this first termine, then the limit of the second term. So have limit of three times and plus one over to N plus A limit s and approaches infinity. Of course, one attempts to U N plus one all over three and squared. So wanna go ahead and distribute my top? So I'm gonna have three and plus three over to n Muslim is an approaches infinity of two, one squared plus three and plus one or three and squared. He really haven't as Anis Tane alone as enemy is approaching infinity. So since going to and bendy are lots of things that we can look at, which top or bottom has the greatest exponents? So here, if two is a great expert to his grace explode. But that was bigger than down here in this same gun. Chimp Indy, The bigger explanations in the denominator it would go to zero. But when they both have the same largest exponents on the variable so here that both Havel one here both have a too the limits just going to equal the fraction of their coefficients. So our answer here is going to be three halves plus two thirds which is equal to thirteen six

Gentlemen, this is a problem. Number 43 from section 5.2. And essentially, the problem is discussing reminds sums, and we want to over some interval, um, forgiven function. And we want Thio. Find a formula for the and three mon some and then take the limit. Okay, so our, um our function here today is the, um is equal to f of X R f of X equals to one minus x squared. Okay. And now, to give you a brief idea of, but we have to do our, um Let's take a look at the graph here, so I'll draw the, um uh, so I'm gonna draw the grab ups. Come on. Okay. So I'm gonna draw the graph of the function here. Looks like it looks like this. Um, so we'll take the interval like this. Um, and, uh, course, this is, um are interval down here from 0 to 1. Um, and our function looks roughly like, um, this here. Okay. Um, no. So what we want to do is we break the on the end. Um, on the end, some Ah, we break the interval into end equivalent, um, sections or ah, absolve the intervals of length one over end eso. In this case, it goes something like this. So, for instance, if we're, um, looking at, ah, the 4th 1 are made Maybe the 5th 1 here, we break this into five different subsets. So this is, um this is won over five hoops, so this is sorry about that. Okay, so we have, um um one fist. Um, this is a ah, boy. Come on. Okay. Um uh, this is two fists, two fists and so on street office and four fists that's so on and so forth. So this is, uh this saree here is, uh, 4/5. And, um, this one in the middle is street office. And, um, then the, uh So you mean, uh and then, um, at each, uh, beach endpoint. What we're doing is we're forming. Um, we're forming a, uh ah. So you you form a box here, Basically a rectangle. Um, and, uh, this here we form rectangles with this, like this, and this s so we're using what they call the right hand endpoint for the boxes. So that means that the height, if you will so are with, of course, is the, um are with here is the 1/5 or with here, of course, is going to be the one over. And, um, and our height here then will be, uh, this This our height, if you will, is, um the, uh, ass off the height is half of in this case, it would be half up 2/5. So it would be our height would be s of, uh, to this. So when you calculate the area of this box so the air will call this box, maybe call the box? Uh, A In this case, it would be, I guess, a two. Okay, So this box here well, just name it as a too. Okay. Now a to, um Izzy s Sorry. The area. Um, yeah. Maybe I should have called it are two instead of a two. Yes, because our two would have been a better letter, I guess Are, too. So the area of our two from the area of and chu, um, is equal to Well, it is half of 2/5. Um, to this, uh, multiplied by the, uh, the with, um, which, of course, is one over. Yeah, and another way, which is kind of the ways that we're using for this, um is the the way that they use in our the way that the reminds some set it up is that this is equal to, um this is equal to, uh yeah, that's ah f x will sit roll call it F k um, times the delta dull Tahir uh, x k and what? Delta what this is referring to. So the delta of X k but it's actually talking about is what is called the increment, um, lengths. So if you notice so in particular here, the increment length is the is the difference between the two end points. So in particular it's, um it would be, in this case, two fists minus 1/5 or otherwise x k minus x k plus one or something like that. And in particular here, reusing the right hand and points. So the right hand endpoints refer to the end point on, I guess the quarter right hand side course, if we're using the left and end point is then, um, the boxes would look a little. The rectangles would look a little different in that instead of the one we have. It would go, Uh, they're rectangles would look like So. So the the area of the given rectangles would be a little different. In that case, obviously, they would be larger and here. But, um, the the idea, at least in these remind sums, is depending on the point to you two, you're going to get a different areas for each some. And, ah, then the, um is the ends Riemann some is given by or these two formula that they that they use in the book looks like this. Um, it's ah ass And, um is equal to, um, the Sigma notation, of course. Ah. Which, um, you know, looks, uh, like this guy. So, um, as, uh, yeah, says, um, Sigma of half of X. Okay, um, times the as I say the delta the dealt er excuse me, Delta X, k and crucially, here, um, the ah, end this and be in this problem. Um, were our increments are gonna be of the same length. So in each case in this problem, um, the Delta X case will, um, be equal to one over end. Okay, So are you on this problem anyhow? And other problems will be a little different, but in this problem, the increments will always be of length one over and And the, um, f x case are always found by the right end point. So, um, are are, But we have to do is first find a closed foot what they call a closed form or s in here, and then use that to calculate the limit. So, um, with this in mind and let's begin. So, uh, first off, um, the ah for you given ends of just an arbitrary number. And, um, we get that r s in here, uh, our s and, um, again the, um Oh, the increments. So since we started, So So it starts at, um, Kay. Since we're using this point are for Kes. Our first rectangle will go from if we replace instead of five here. If we replace this by end, Goethe one over and you over. And, um, you know, dot, dot dot This would be maybe instead of three over. And, um, this would be ah, Villa A. This would be at three over, and and then, you know, one, of course, is is is, um is an over end. Okay, so the one before it would be Ah, um ah. Let's just say how do you want asses and minus one over and yeah, it's getting a little tight here, Um, but and minus one over in and ah, and minus two over and all the way down to ah, and then three over and two over. And so B s. And, um So the first thing we should try and figure out is what is our, um what does our index k go over eso in this case, Kay goes from, uh, que goes from one because we're using this side, remember? So it's k equals toe one. Um, k equals to 12 And like I said, uh, it ends at one. So in this case, it's K equals one, two, um, in here So it's K equals one to end. Um, and then the f x k refers to the this right hand endpoint. So that would be, um it would be f off. It would be af off, uh, K over end and then times the length, which, of course, is, uh, one over. And so I'm gonna simplify this now just to eliminate it like this I'm gonna just write it as af of K over and, um, divided by Yes. Okay. It's simple. Find that here. Ah. And now, in this case, of course, the function I'm using is one minus x over two. So I'm just going to replace af of axe over and buy one minus X over too. Um so let me just do that here in this form. You here. So I'm just gonna replace this by, um, this becomes, uh, one minus. Uh, Now it's k over. And so it's, uh, que over and square. Okay. And, uh, okay. And now we want to do, of course, is continue to simplify this now. Ah, so you'll notice that your summing from one end. So if you're something, um, from want and ones, then you're going to get in there. But then you're dividing by ends. You're going to get one. So this is equal to, um this is equal to one and then minus um uh, is this some? That's some again here. So it's going to be minus whips. Uh, Minus again? Yeah. Is the, uh, um K squared? Uh, we're gonna get troops, huh? K squared now. Provided by and square times end gives us it's divided by n to the herd. Um, provided by, uh, case by the end square times end, which is one over our wishes and cube. And again, Now, the sum goes from K equals one. Uh, so this some goes from I want to end here. So this is our some and, um, Eagles want to end. So now we're now we've calculated Are we calculate ups. Okay, so we've calculated this is ah, um, this is our least our first form. You. LF you Well, this is our first formula for for, uh and you're sorry for s And so this will label as maybe, uh, f one. Okay, so we'll call this F one. Okay, so now with f one in mind, um, now we have to move forward here. Uh, we can't. We need to use a formula. There's a nice formula to make this easier. So So we moved to the next stage, and, um, we want to find a formula for the end. Okay? Eso We use the formulas that, uh, this guy here, which is, um uh okay, So we used a this formula, which is actually in your book, So, um, but I'm just repeating it Here, Um, this formula here it says that the squares So the, um a equals one. Ah, to end here. Ah, wake up. Come on. Sometimes it's gonna be so frustrating. Okay. S O okay. I'm sorry. I see what the problem is. It's too close to the top here. So in other words, it goes from K equals one hand. I'm sorry. It's just too close to the top. K equals one end of this guy is equal to this end and plus one time to n plus 1/6. So we use that. So then, um, from F one. So from f one, we get, um, from F one, we get that, uh, s and is an equal to one. Um, minus. Now, remember, we have. So, uh, we have this, um uh, the the F one. Oops. Uh, So the F one was this case, This some divided by n cubed. So maybe maybe what I should do is actually factor out this and cube here. Ah, see? So if I factor out the end cube, uh, you'll see that this some here is actually the, uh, and shoot. Um, so this sum and now it's one minus the sky. So it's so this here we have from, uh, this This formula here gives us one minus. So, um, it's one minus this. See, in n plus one, sometimes two and plus one. Ah. And now it's divided by the six times the end cube from the formula week from what we had before. So this is then six thistles. Six and to the third. Okay. Ah. And now, um, what we can do is we can expand this out. Well, actually, I prefer to just cut to just cancel the end here, and, um And that would make the denominator ace, too. So instead of that, I get and squared. Um, And now what I get is that this is then equal to one. Um, minus. I'm gonna write out school. I'm gonna use the foil method to compute the numerator of that fraction there. So I get, um, to and squared, uh, plus three and plus one. And, of course, it's now divided by, uh, six and square. So we get here, um, six and square, dividing each, uh, each fraction. Uh, I'll be just sort of standing the given fractions here. Uh, me. And so what does this, Uh, what does this tell us? Well, we have then is ah, we can go through then. And in particular, I want to simplify this down, so you'll notice that the end squares cancel in this 1st 1 So I lose the end squares here, and this becomes one over free in the 2nd 1 The end cancels with this one, and I get Well, it doesn't matter here, but the final one is just 1/6 and square, so I can simplify this down further. One minus 1/3. Ah. Then gives me, uh Well, let's let's just move on to the next one. So So one minus 1/3 would give me 2/3 here and then plus this of their stuff. So, in other words, I get, um that my s and by s and eyes equal to, um you see, it's equal to 2/3. Two over three. Plus, these other guys, which are, um, or I should say, minus one over, uh, to end. And then it should be minus. Sorry. Um um, minus one over to end and then, um, minus 1/6 n squared 1/6 and squared. And so this is the formula for my s end for my aunt. Uh, sorry. My ends reminds some. OK, And now what I want to do is just compute the limits. In other words. Now what I want to do is just look at the limit. Um, as an approaches infinity, um, of s and and, of course, uh, this is, uh, is equal to, um the limit is end goes to infinity of that, you know, some above, which is to over three minus one over to end. Um, minus one over six and two. The second. Ah. And now, hopefully, you know your limits Well enough to know that this these two terms here, both of these two terms are gonna go to zero. So this is going to go to zero. And this will also go to zero. And what we're end up. What we end up with is then, um, just to third. So we end up with 2/3, and that is our answer. Um, eyes 2/3. And that's it for Ah, this problem, folks, it's a bit of a long, long problem. It's, ah, number of steps, but, uh, that's it. Uh, thank you very much, folks. And have a nice day.


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