a-estimate-the-value-of-lim-_x-rightarrow-inftyleftx-sqrtx2xright-by-graphing-fxx-sqrtx2x-over-a-sui

Question

a. Estimate the value of 
$$\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+x}\right)$$
by graphing $f(x)=x-\sqrt{x^{2}+x}$ over a suitably large
interval of $x$ -values.
b. Now confirm your estimate by finding the limit with
l'Hôpital's Rule. As the first step, multiply $f(x)$ by the fraction $\left(x+\sqrt{x^{2}+x}\right) /\left(x+\sqrt{x^{2}+x}\right)$ and simplify the new
numerator.

Doruk Isik · Accepted Answer

in this problem, we are given limit off function where we&#x27;re gonna pull this election bikes. And we are first as the a graft function. You could use any. Oh, my function off. Porter, if you do that, you see that as extra. See infinity. So this is the XX. Is this functions now? The approaches to negative point. So then the limit from McGrath. You see, that movie was a big war house now. Important. This was part of a part feet we&#x27;re gonna cut with this year&#x27;s Eve. Look no true politicas except purchase it differently. And actually, we&#x27;re given a hint we our asses to multiply both writer and intimidated by explosive period of expert. Plus you&#x27;re right by X plus spirit of spirit. Plus thanks. The in the numerator people have limit is exposed to infinity. You don&#x27;t have explained minus expiry place. Uh, X divided by now X in the Ninomiya is account from protector. So we can write this with this. Um thanks. One plus scrape bird one waas one over X. Now these will go away. X will cancel out, but on the limit will become limited&#x27;s extra 50 50 um native one divided by one WAAS spirit of Warn plus four X as except purchase. See pretty this term we&#x27;ll go to zeros and then this will be negative one over. Well, today&#x27;s will be negative one, so it isn&#x27;t safe.

Ahmed Ibrahim · Answer

in this question. We have two reports. First Part&#x27;s asking toe Throw a graph off this function. Why equals X minus square root X squared plus X. So if we and if we substitute by X, equals zero, we get very why which will be equal to zero toe. So starting from zero, and by increasing value, we can find it and go as trend off something like these. And by increasing value of X, it almost reach to a negative off. Honest, a constant. So the second party in these question asking toe verify this graph using limit when X approaches infinity so we will start by multiplying and these equations or dysfunction by the same function was different. Sign. So, for example, X plus square root X square plus X divided by the same, so you will not change anything. Excess square lost X now, so the result will be equal. X square Linus X squared plus X What Eggs lost square root X x square, plus thanks. Now we can remove or simplify. X squared is X squared off the negative, so our present with the equal X or negative eggs, eggs plus square root actually square classics. Now we can take a drive it off. So we will have negative one over one loss. Square root and under the square root. We drifted off ex queries. One on drifted off X is one over X. Now, if we substitute by X equal toe infinity So this turn will be equal to zero because one over guarantees you. So our final answer will be negative. One one last square. Root for me. Cheese one. So it would be negative. 1/2, which is seen as Barty. We sold before Biograph. Thank you.

Daniel Jaimes · Answer

This is problem number forty five of the Stuart Calculus Eighth addition Section two point six party estimate. The value of the limit is experts and I get of affinity of the square root of the quantity X squared plus x plus one plus x. My graph in the function F of X is equal to on the square root of the quantity exportable sex plus one complice X. So my graphing here we can see the grass of the function. Oh, but notable. Ah, point here towards very large negative number showing that the number is approximately negative zero point four nine three and the number It was slightly decreasing more and more as we went along. S so this is very close to about native one half. So the best estimate that we can get right now isn&#x27;t this limit is approximately equal to negative one hour or is there a point five parties the table probably use of effects against the value of the limit. So if we were calling table values here and we see that we a cz, we increased the value the value of X to a more negative in negative, larger number at the value of the function itself perches Native Sierra Point fire. Until at this point, teams have equaled it. But of course, we&#x27;re approaching the surprise. I haven&#x27;t never reaching it, but we were able to confirm that are estimated from party is fairly accurate. And in part c, we&#x27;re gonna prove that this estimate is absolutely correct. Analytically. So we&#x27;re going to go ahead and workwith this limit here first going to multiply by the contra git to simplify this function a little further. So it&#x27;s this quantity minus X divided by same quantity. Right More to plane by the contra. Get to the top and the bottom. This will give us it&#x27;s limit. Tex Purchase negative infinity of Ah, this quantity X squared six. That&#x27;s one my Mr X squared. That&#x27;s what we get When we multiplied by the country and the denominator we saw the same term X squared plus x plus one Ah! And minus six. Okay. And the numerator We see that an X squared term can counsel and we&#x27;re left with just experts. One. Want a little more supplication? Still, the limited sex purchase Negative Infinity, um generators X plus one in the denominator we&#x27;re going to do is we&#x27;re going to factor out and x squared from this radical. Which leaves us with one bus. I wanna rex this one over X squared. It&#x27;s a little bigger. One plus one over X. This one Red X squared and then I&#x27;m in sex on the outside. Okay, Regarding this square root of X term since we&#x27;re in the negative domain, we&#x27;re going to take the following steps. The square root of X squared is equal to the upside Volume X And because excess listen zero we use a replacement. That devalue absolutely of X is equal to negative X again on ly in this region of our interest as experts Negative infinity. So we make that substitution Women his expressions negative infinity out of this quantity X plus one departed by negative X multiplied by this fraction or this radical or in plus one of Rex This one over x worried minus X. And our next step will be to divide by X so limited sex approaches. Thank you to infinity. Thanks a lot. About except one one of my exes. When Rex the radical that nominator doesn&#x27;t change because this x is cancelled that just has a negative in the front. And then we have made a sextet, Rex, which is one and when exports is negative. Infinity dis fractions on Iraq&#x27;s upper to Syria. Wanna wreck squared? Also purchase here and so we&#x27;re left with is one place here in the numerator or one negative squirt of one minus one in the denominator which simplifies to make it a one or two or negative one, which confirms our estimates from parts and me.

Yaw Asomani · Answer

so risk etching FX equals screwed off X squared plus X plus one and then plus X nightclothes X is not There&#x27;s no containing his great food sign. Right? This one is not contained in his grave site. And this is a rough sketch of it is down trying to escape Once again, you can use a graphing calculator to see the real thing, right? Eso you have this one right? And you gonna, uh, estimate the limits of those function as X approaches. Negative infinity, Right? This is how except which is negative. Infinity. It goes like that, right? Negative one negative to negative for negative five. Seclusion. Negative infinity. What is happening to the curve? Look at a curve. We respected away access. When you trace the current well respected it. Why access to concede that the limit off those function as exit pushes naked infinity Its negative soup on five. Right now I want to use tables can use tables, right? Can you stable to know how to use tables? Right? Eso you can make some table. This is X into cell affects reusing dysfunction Whoever you see X put, uh, negative 10 right When it put negative. 10. What are you gonna get? You&#x27;re gonna get negative zero point forces Sierra six year negative. 0.4 six years. 60 Right when you put 100. So using ah table form to find it right as X approaches infinity. So it&#x27;s gonna start from 1,000,000. Negative. One million is going to negative infinity. Right? So when you put negative 100 what you have, you have, uh, negative. 4.962 right, Negative. 0.496 to you know, a lot of things, you know. So you do. Negative 10,000. What is negative thing? 10,000 is negative. 4.9996 negative. 0.49996 You know, So it&#x27;s getting closer to negative soup 15 just like we said. Here. Photograph. Still we see that this for the using tables. You can see that Still the limit as X approaches infinity off. The function is approximately negative. Negative 0.5 right can see that it is always getting closer to, uh, negative 0.5. Right. So that&#x27;s business. When he used tables, you&#x27;re gonna arrive with the same answer and when you use, I&#x27;ll do you break methods. Right? Limit. Except for which is negative. Infinity off X squared. Plus expose one, then plus X right. What do you do? You mortified by the congregation? Right, So this is gonna be X squared, plus X plus one minus x forward. Same thing X squared plus X plus one minus. Uh, ex. Right. This is the congregation. Then what is happening? The numerator is gonna be the first term, Which is this one squared, and that is just a term under its were sign. Right? And mine is the 2nd 1 square, just the 2nd 17 term squared, right? And then you&#x27;re gonna have over, uh, x squared plus X right plus one, then minus x like that. Right? So finally, you haven&#x27;t expose one over square root off X squared plus X plus, one minus x, and we&#x27;ll find a limit. Asked approaches. Negative. Infinity. Right. And then what is happening? Uh, see here I can do something. Manipulations. I want to manipulate what is here. I can factor out x squared. When I do that, what am I gonna get gonna give one plus right one over x plus one over X squared, right? And then my sex. I can do this. I just factored out X squared in this What is under the square root here. And then when this x squared come out is gonna be x plus one over, uh, and they have ah. One plus one over X plus one over X squared. Like that. Right. This is great route here. Uh, you have this one, right? Um and then you have minus X here on a denominator. Right? So I factored out x squared with a square root out. What is the square root of X, where it&#x27;s just x right. It&#x27;s just X, But because of the limits as X approaches negative infinity, you can just see X. You have to say absolute value of X ray because of the definition of the absolute value, that is gonna be absolute value of thanks. And then this is gonna be one close one of her ex post one over X squared minus X And then on the top, you have X plus one and you&#x27;re trying to find the you limit as X approaches. Negative. Infinity, right. Limit as X approaches negative in free. You can just take away this group like that because of the negative infinity. You don&#x27;t have even a negative square root. So once you have this one now, what is the definition of the absolute? By definition of absolute values that going exits less than zero. Then it is negative X and one excreted and zero. Then it&#x27;s positive, X right. And it says exits approaching negative. Infinity means that exists negative. Therefore, you take away the absolute value sign. But you put a negative X right, because excess approaching negative infamy. So you have here limits execution Negative infinity and then just square, which is gonna come here one cause one over X plus one over X squared minus that. And this is one X plus one. Now, what is the highest power at? The denominator is one. So the divide each and every side you can every turn by one, right by ex big important. The house power off the denominator is extra power one, this one. Right. So I divided it at every turn by extra power. One exit a power one is what is gonna be exit a power one right? Then This is one plus one over eggs Close one over X squared and minus X cooper. Excess. Well, right. Anybody can return by X. So what is happening? And this becomes a limit. As X goes to negative, infinity has becomes one cause one over X, then over. Negative. Swear Route one plus one over X plus one over. X squared minus one. Right. So if now exposed to negative infinity, this is gonna go to zero. This is gonna go to zero. So, what is happening? You&#x27;re gonna have negative square root of one, which is, um, negative one. Right. So here, this is gonna be negatives. Grab it off. One. This is gonna go to zero. So you just have one of the numerator, right? This one here and then over. This is negative Square root off one, right? What is never discredit. Screaming off one. It is negative of one. And then this negative 12 He&#x27;s gonna come here. So is negative. 1/2 right, which is the same as negative 0.5. So, by algebraic manipulations, we&#x27;ve seen that a limit as X goes to negative. Infinity is negative. 0.5 and then by tables we saw that the limits as X goes suit negative. Infinity is also negative. 0.5 and then by graph we saw that in limit asked as goes to infinity Negative infinity. It&#x27;s also negative 0.5 right, So they all point to seem

Yaw Asomani · Answer

please. If you didn&#x27;t watch the last two tutorial before those one I urge you. I strongly urge you to watch it, right, cause it&#x27;s gonna be very important for you to understand this one, right? Especially the last tutorial. Right. So, uh, we have those thing right here. Right? So we have this and then, um so this one is gonna be, uh, X squared, plus B X. Right. This is supposed to be minus so like we did in the last two. Terrible for this one. We&#x27;re gonna multiply about it, congregate. Great. So this becomes the limit. We can forget about a limit at a moment. This do some manipulations first. After redundant of manipulations. Weaken. Don&#x27;t take the limit. Right. So do not be concerned if I do not write a limit at some point, I just I just wouldn&#x27;t want us to finish the manipulations first. And then we can we can We can think about the limit later, right? When I put it in a suitable form, and then we cannot take the limit. Right. So this is gonna be we&#x27;re taking taken the conjugation, right, So to congregate. It&#x27;s just the same thing but we just changing the sign from negative to positive, right? That is how we do conjugation. So this is gonna be, um X. Right? And then posts were off X squared minus So close. B X right to have this one. Uh, we know that we multiply it. Such a thing here like that. The numerator is gonna be the first term square. This is the first term. If you square, it is just gonna be a term on 30 square Resigned, right? Fine is the second time. So if you&#x27;re square, this one too is gonna be the one under the square root sign right here. Right. So that was the numerator right in the denominator is gonna be Ah, same thing here, right? That is the denominator. And so this is gonna be X squared, plus B X. Great. So once you have that, all you have to do now is do some manipulations. Right? So you know you have this one this minus this one. What is it gonna be? You know you have Ah, you have X squared here minus X quit is gonna go to zero. So all that you have is a X minus B X. Right? And so it is gonna be eggs excess common. So I could bring X out that I have a minus bi, right? Ah, so once I have that, what is happening? Or let me just Yeah, let&#x27;s this a minus bi And then what is happening at a denominated? What is I&#x27;ve been a denominator. Is that? No, I&#x27;m gonna, uh I&#x27;m gonna What is the what is the highest value between Thies too? But what is the highest power off X in in here? Right, You know that it is not x squared, because off the square root, right, it&#x27;s not excluded because of the square, but it&#x27;s actually X right? So if it is eggs and I&#x27;m gonna divide each and every turn here but the numerator and a denominator by experts, that is gonna be X a minus bi over x. And then you know how we did it. The last tutorial, right? So I&#x27;m gonna do the same thing straight away, right? So this is gonna be you ex Quit that I said if you didn&#x27;t watch the last tutorial, please do it right, Because this is explained there. Right? So this has been a BB X over X squared us well, and it&#x27;s still divided by X because we&#x27;ve just square root taste. This X squared is gonna be X right. That is still so. It&#x27;s still divided, divided each and every side by eggs. That is why I said watch that last two tutorials explain it there as well, right? This is canceling this one. So this is gonna be x A minus bi, and then this is gonna be now, this would take on this one, cause you&#x27;re gonna share, right? You&#x27;re sharing the new bridges over the denominators. So this is gonna be x squared over a squared and plus a X over X, and they actually write it X squared over X squared, plus a X over X squared, Right. That&#x27;s what this wedding up, I mean, right, So once you have this one, then the next one is gonna be the next term is gonna be plus also square root, X squared, minus X squared, plus B X over over X squared. Right. So now this cancels this and indistinct that we wanted thes. And then there&#x27;s cancer. There&#x27;s an indistinct way one of us so What you&#x27;re gonna have finally is is a minus, bi over. Screw what is here? Here is one. And in here is eight x x ray. So plus a over X. And then plus what is here? What is here is also one, then plus B over X ray. Now we continue limit as executions. Infinity. So if you take the limit as X approaches infinity, what is happening? Uh, this one is gonna eventually go to zero, and this is in a eventually got a super because A and B are fixed constants, right? So if a and B are fixed constants, then they&#x27;re gonna go to zero. So, are you having a denominator? What do you have? A numerator is a minus. Bi, right? What? Do you have a denominated? Not matter. Here. You have screaming off one plus squeal off one. What is good of one is one. So it&#x27;s gonna be actually one plus one, which is two. Right? So this is what you have as a final

Yaw Asomani · Answer

Now, if you grab this function, um, if you grab dysfunction if you graft dysfunction right in front here, uh, it takes a format that is like this Just using a graphing calculator, right? Alu online And use any graphing calculator you confined to the form that that And then this point here is 0.0.667 Right, So this point here is the coordinates. Here is zero for the X at one point 0.667 Right. When using a graphing calculator 667 So this one is actually approximately. When you when you pluck this one graphic on the graphing calculator on the internet, this one is approximately zero point. Um 0667 That is the y value, right? This is your disease, The x coordinate. And this is a white Cornick, right? So So that is that is what you have. Eso No, we want to do. Ah, a little table table of values. So when you have eggs, what is the correspondent effort face, right? I mean, you can start from any place. I&#x27;m just gonna start a little closer at u zero c negative 0.1 right? What is it, Chris found in, uh, function body function about it. Just put negative 0.0 Israel one here. And you&#x27;re gonna get a correspondent. Functional value of, um um, 0.6661 So, 0.6661 ah, 0.6661 66 zero, point 6661663663 Uh, then you go a little closer to zero by adding 1/3 0 So this is closer to zero than this one, right? The 2nd 1 is close to zero in the 1st 1 so that when he says we&#x27;re gonna be the same thing 0.6666 this time you have four sixes, right? Ah, then 617 angle most step further to zero. It&#x27;s a step closer to zero by adding 1/4 0 So four zeros here. Right? So what is that gonna be? That is they&#x27;re gonna be, um I mean, almost negligible difference. Right? Still gonna be 0.6666 Right. Um, seven. So it&#x27;s just calculated work right now. Move to the positive side. The positive side is gonna be zero point 00001 with four zeros. Right, cause and I&#x27;m moving to a positive point. And is there gonna be the same here? Uh, zoom 0.666 Um, 67 This one is actually five sixes and several 17 See? So it&#x27;s just I&#x27;m gonna meet one more value. Uh, this produce this ones are moving further away from zeros are reduced the zeros by one still zero point. Um, zero point. Um, say 670.6667 Uh, 167 So, 167 So you can see that as you getting closer to zero. This is a functional value, right? Is the function about you. So, what is this one? This is actually a fraction, right? This is approximately two of the three, right? Says approximately two or three. So, as you approaching zero, you can see that the function is going to 0.666667 That is approximately two or three, right? These do can be approximated to to over three. So the limits as X approaches zero off. This one is approximately two or three, right? This is the 1st 1 Eso eso once you have that, uh, you want to know, uh, finally, estimate the limit itself without having to, uh, use it table, right? We&#x27;re gonna gonna do what we&#x27;ve been doing throughout. Right? So what have you been doing throughout? Well, we&#x27;re doing limits. Ex approach zero of X over. Screwed of one plus three X minus one. Right. If it put X acquits demonstrated where you&#x27;re gonna have zero of zero. Right? So you see, it&#x27;s an indeterminate for so you have to do more work and what we&#x27;re gonna do, we gonna multiply about a contract. That is what I said and meant by saying would have been doing all this while, right? So finding a limit, aren&#x27;t you? Briefly. Right. So what is this gonna be? This is gonna be X times one plus three x plus one, right. And then the denominator is gonna be the first term square minus a second turn square. A new that So one cause of this square is Is this a 2nd 1? Which is this term here? One square is also one. Right. So you have this one now? One minus one is zero. So Oh, that you have is three X lettuce left, right. And this three ex this ex is gonna cancel this X. So you&#x27;re actually gonna have limits X zero off school of one plus three X right? And then plus one here over three. Right now, when you put X equals zero, then what are you gonna have? What you gonna have is screwed of one, which is one that one plus one is two. So it&#x27;s gonna be to over three. So you see that we get the same answer as using this table here, right? Get the same answer. But this one, we just used tables and programming values to get our Ju three an approximate fashion. But here you do it, Archer Brickley, ever get to over three.

Problem

Find $\lim _{x \rightarrow \infty}\left(\sqrt{x^{…

Problem 81 Medium Difficulty

Answer

a. $-\frac{1}{2}$
b. $-\frac{1}{2}$

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a. $-\frac{1}{2}$b. $-\frac{1}{2}$

Thomas Calculus

Grace H.

Catherine R.

Anna Marie V.

Michael J.

Precalculus Review - Intro

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George B. Thomas Jr.Thomas Calculus

Grace H.

Catherine R.

Anna Marie V.

Michael J.

a. $-\frac{1}{2}$
b. $-\frac{1}{2}$

George B. Thomas Jr.

Thomas Calculus