a-use-a-graph-of-fxleft1-frac2xrightx-to-estimate-the-value-of-lim-_x-rightarrow-infty-fx-correct--2

Question

(a) Use a graph of
$$f(x)=\left(1-\frac{2}{x}\right)^{x}$$
to estimate the value of $\lim _{x \rightarrow \infty} f(x)$ correct to two decimal places.
(b) Use a table of values of $f(x)$ to estimate the limit to four decimal places.

Yaw Asomani · Accepted Answer

Susie function is just part of a function. Uh, FX is equal. Teoh, Uh, one Y minus two over X to the Power X. This is just part of it, right? There&#x27;s another part like that. But I&#x27;m not interested in this one because I&#x27;m looking at the limit as X approaches infinity. And this is the part that we&#x27;re looking at because extrapolation infinity is like this part, right? So that is why I did not concern myself, but he had a part like that, right? So I&#x27;m just concerned myself with a right side because of those love it right as X approaches infinity of this function. Right? So what is the limit as X approaches of remedial the function, um, one minus two over X to the power X year, right? What is what is that limit? So when you zoom in your calculator, you can use your graphing calculator or you could use on online graphing calculator right under Internet. When you zoom any and you can see that here, you trace it to the white value. Consider here is approximately what, 0.135 right? Because it is at this point that the current levels out. Curious level in is almost horizontal here. Right. So if you zoom in your calculator, what do you use an online calculator or using a graphing calculator yourself? You can see that it&#x27;s approximately 11 and out at 0.135 Right, Because this is super in one, right? And 0.15 So 0.135 is somewhere in between, right? So when assuming probably can see that this momentous approximately 0.135 Right. And we can also do finding by the table for right every face you can plug in values of X into this function to see where it is going. Right. So when you plug it into a table for what you can start from 10 you know 1000 you know, 100,000 and all that can go on and on because you&#x27;re approaching infinity, Right? Richard Infinity here. So you could go on and on and on and on. And each time you do that, say, if you do 10. This is Sue 100.1, uh, 07 073 right. Uh, seven. You know, if you do 1000 right, it could continue in 2000 1000 is number 13500.1350 Right. And then if you get get to like, ah 100,000 you still get it. 0.1353 threes. He&#x27;d so you see, it is still in the neighborhood of 0.135 Right? So you can see that as you go on and on and on, you get in something that is in the neighborhood of 0.135 approximately. Right. So if we use a table, gives you that firefighters 120,000 1 million do continuously. You can see that you be getting in the neighborhood of 0.135 Right? So therefore, by the tables, you could also see that the limit The limit here is true, right? So the graph in a table uses the same arrives

Paul George · Answer

everybody we&#x27;re doing Chapter two, Section four Problem 26. So we&#x27;re kind of handling this with a little bit of knowledge of like what we used to do in the past sections mixed with our new way to do it. So it says, Hey, just, you know, look at this graph, right? We have squared of three plus X minus the square three. We divide it. All of that by X. We&#x27;re trying to find the limit when X approaches zero, right? The problem is that obviously you divide by zero people again. Zero. So that&#x27;s a no. But, um, we&#x27;re doing this for the difference question for this whole thing. We can just kind of look a graph, right? And so if you pull up, does most or whatever you want to use, then you find that this is like point Thio. That&#x27;s my 0.0.3 or whatever, right? And you have this graph. It kind of comes in and it goes like that. And that kind of just keeps going. And so you can kind of, like, eyeball it and get it to about, like point, you know, about like 0.28 right? That&#x27;s what the first question what you do it is. Hey, look, a graph and kind of approximated toe to desolate place right on. Then it says, Hey, do you know that was That was party, right? Hey, look, a graft and kind of get kind of get close right on. Then The next one says, Now, we want you to get a little bit more specific, right? We want you to get a little bit better of an answer. We want you to get to four different places so you can start plugging stuff in, right? You can say, Hey, I&#x27;m gonna go it. Zero It&#x27;s gonna be something. Right? And I wouldn&#x27;t get kind of close. So maybe I&#x27;ll plug in, You know, like negative point. 05 Right. Maybe you plug in like a negative point. 01 that kind of thing, right? You get, like, super close and you end up like I got, like, 10.28 And so if you put him in kind of far away, you get like 0.29 and then you get closer, you get like 0.289 and then you get, like, a little bit closer, a plug that one right there. It&#x27;s like 10.28 going to a 87 Right. We&#x27;re getting, like, a little bit more specific on dhe. That&#x27;s sort of like the whole goal, right? Is that you want to get a little bit closer. You go. Okay, Well, that&#x27;s what that is. And then they say, Okay, throw those methods away. Those air close, but they&#x27;re not exact. I want you to give me exactly what this is. I want you to use limit laws I want you to use with the section taught you. So that is ah lot of what we&#x27;ve been doing before. So now we just have to figure out how in the heck did we make it so that the top has a factor acts to cancel the factor of acts in the bottom. The answer is contra gets. That&#x27;s at least how I&#x27;m gonna show you how to do this thing. So just find the contra, get to the top, which is actually to con tickets. But I&#x27;m gonna choose the one that seems a little bit more obvious to me. Take the continent of the second part. So we got plus square to three and then divide that by and I&#x27;m just going to say the same things. I don&#x27;t want to write it again. So then you just have the first part. That&#x27;s the nice things squared, Uh, congregates cia three plus x and you get minus three. And then that&#x27;s over x times that congregate thing that I don&#x27;t want to write again. And then you see that the threes cancel the exes, they divide each other, so you just get one over. And now I will write the contra gets three plus X and remember, the whole goal of it was to get rid of that ex right, Because that made it so that we divided by zero. We got rid of it by doing some arithmetic. And if this wasn&#x27;t your first, you no idea was doing congregants. That&#x27;s okay. Try other stuff, go down a path and see if it works and see if it doesn&#x27;t work. I&#x27;m not saying this is the only way. I&#x27;m just saying this is how I do it. And so now if I plug in zero to this, I get square 23 plus zero to square 23 and then I have scored three plus quarter three, which is two square three. So I get one over two times the square to three, which is the exact answer. There&#x27;s no decimals. The square to three is irrational, right? We approximated to a decimal. We are wrong. It says, Hey, tell me exactly what this thing is. And also you aren&#x27;t one over to Times Square to three. Don&#x27;t. And that would be the exact answer. Okay, I hope this help until next time.

Yaw Asomani · Answer

So when we have here, um, And see, we have every fakes. Um Thanks. Right. And then minus. That&#x27;s cooler. Uh, ex. Now it turns catch dysfunction. You can use any graphing calculator, right? Turness case dysfunction is gonna behave this week, right? Is gonna behave this way. Uh, that. And then here you&#x27;re gonna have the coordinate at here. You&#x27;re gonna have the coordinate of zero and point to night. So is zero and 00.29 Right. So you can see that, uh, ever effects, which is a wide value, right? It&#x27;s 0.29 This is the Y value that is there for Frank&#x27;s Vatican. Right. So, uh, this one, this one is approximately 10.29 Did you know point to night every face when you try to find a limit of that as X approaches? I suppose she zero great with the graph. Does this point to night now, in a time finding with a table A and try to find a table trying to find bug in values to see it. Approximate values. Uh, what are you gonna get? Okay, So what you gonna get? Let me start from something that is committed. You&#x27;re like, uh, point negative. 0.1 Right. So when I try to start from, so this is gonna be my eggs value, and there&#x27;s no being my app effects. Right? So when exits negative 0.1 Right. When X is that, then what do you have? You have 0.2886 uh, 992 And then we decrease it further by adding one more zero like that. Uh, have 0.28 86775 So 867750.2886775 Now we decrease it further. So this is getting closer to zero as we add more zeros here. Right? So this is closer to zero. So, uh, it&#x27;s cause it zero from the left, right from the negative side, that is 0.2886754 Right. So this is point 2886754 I believe, uh, yeah, check my cackle in. And then now this jump to the positive side. So this is a positive. So this is also closer is also closest. Zero from the right. Right? So when you do that, that is also point to a 674 night. Right. So this is 0.2886749 Now, let&#x27;s go, Father. A little bit by reducing two zeros to three. Right? So we do reduce the zeros to three. What is this one is. Oh, so, um, point to a 6727 praise so 67270.2886727 You know, so we can continue to decrease it further and further from it. So this is the furthers right from the positive and a negative. So where is it approaching that? It&#x27;s approaching 0.28867 Right, So it&#x27;s a promotion. This one is approaching point to a 867 Right, 0.28867 here. So when you use tables, this is waiter. Uh, limit is approaching. Right When you&#x27;re around those one it is this soup point. Nice. Well, right now, let&#x27;s do this algae, Brickley. Right. So we&#x27;re gonna do this one ology freakley now, So you gonna find a limits? X approaches zero of the function, right? So what is the function gonna do it? Uh, actually, Really? So this is gonna be a three plus X minus over minus minus three over X. Right, So it&#x27;s minus three over x years of minus square, root of three. Over X. Right. So what we gonna do when put zero there? We&#x27;re having, uh, indeterminate race. So we&#x27;re just gonna continue with our motives. Operandi congregates, right. Remember? So the multiplication of the top one is gonna be the first term square screaming. That&#x27;s great of that is just this right Then minus the second term square, which is also this and then the denominator is just gonna be the multiplication of that. Right. Plus square root of three now three minus 30 Right. So you just have X and this excess cancer in this X. So what you have in your hands is the limits as X approaches zero of one over squeal of three plus x plus screwed of three. Now put X equals zero and you put X equals zero. What you have is one over square or three plus square root of three, right? That is twice screwed of three. So it&#x27;s actually gonna be won over twice. Screw with three, right. You can leave your answer like that or you can try, toe, um, put it in a proper form. Right? Because, uh, radicals or sirs are not supposed to be. Technically, I&#x27;m not supposed to be at the bottom, right, so you can do it. And then this is also the same s, Mr Senator. We have put in it any Which way works, right?

Foster Wisusik · Answer

this question wants us to estimate this limit, then prove our result. Outbreak Lee. So if we look at our graph here at zero, it looks like it&#x27;s between 0.3 and point to a and almost smack dab in the middle. So we&#x27;re just going to say this is approximately 0.29 based on the decimals graph. Then he wants us to use the function values to create a little table. And if he looked, if you plug in 0.1 that&#x27;s approximately the same thing as when you plug in 0.1 in the negative direction. Both ease air around 0.28 nine. So not too far off from our guests. But then, for the actual result, we&#x27;ll go back to our normal techniques for finding limits. So we have square root three plus acts minus Route three. Divide by X, and we have square roots here. So let&#x27;s just multiply by the conjugated like we&#x27;ve been doing. So we have Route three plus X minus route three times. The conjugated, which is re three plus X plus Route three, divided by X Times Route three plus X plus Her three. So now let&#x27;s just evaluate the top, which is just the first term squared, minus the second term squared. And then we&#x27;re not going to touch the bottom for now, and we see that these three is cancel, so we&#x27;re just left with X on top, divided by X Times Square Root three plus X plus her three. So then the X&#x27;s cancel, and we&#x27;re finally left with something we can plug into because zero goes away. So now it&#x27;s just one over square root three plus X clustering three or plugging in zero one plus route three plus for three or 1/2 square root three, which is indeed approximately 0.289 and that&#x27;s our exact answer.

Daniel Jaimes · Answer

this problem Number thirty four was this tour Calculus Safe Edition, section two point three Pretty use a graft. Half of X equals this corbett of three plus six minus a square to three wrecks to estimate the value How the limit is X purchase here over this function to do it to test in places simply plant dysfunction and as is foreign and we see that it approach is very close to zero point two nine. So this limit and party we will apart that parks me for it to be a hero. Two nine party. He&#x27;s a table of values of F Testament limited for distant places. Oh, wait. Use the table to plot this function very close to zero and we see that closer than we got two zero from the left and the right, of course, are we approximated the function value to be point two eight. What happened? Two. Four decimal places and the letter there knew it. Improved estimate party use element lab laws to find the exact value of elements. So where you will really write this as a limit on the next page. Limited expert zero of screwed of three sex Explain over minus the square root of And we will take the next step of rationalizing the numerator. Yeah, by multiplying by square root and Papa&#x27;s X plus the square root of three to the temple about him, which could thiss three plus sex minus three in the numerator divided by X Time&#x27;s a quality square root of the quantity three point six, plus a spurt of three we see here at the three minus three, it cancels out to zero, and then we&#x27;re left with her. One acts in the top, which can cancel with the one X and that there no need her leaving us with no limit his expressions. He around, uh, one over in the square root of the quantity three parsecs plus square root of three. And if we evaluate us at X equals zero, that should give us and exact value one over this herd of three plus a screw to three or one over two times the square root of three. And this is our exact solution. This product are this questioned is equal to approximately on zero point two eight eight seven, which is consistent with our yes, um, it&#x27;s, um, ists from parts A and B

Yaw Asomani · Answer

Sooners have f X equals, you know, screw off. Three X squared plus eight experts six and then, uh, minus Screwed or three X squared, Uh, plus X with three experts world. Right, that is have this one. Okay, so once we have this one, this is the sketch off those function, right? A graph you see in this schedule dysfunction. Once again, if you put it in the graphing calculator, you&#x27;re gonna get it Better visual graphics that officials than mine. Right? This is just a sketch. It&#x27;s not going to scares. Just I just did it using my hand. So what is the limit? As, um, X approaches Infinity, right? X approaches infinity here. So the limit as X approaches infinity. Well, this is the X value. The horizontal. Right. So you start from 1234 going. So as you approach infinity like that as you go to the right off those horizontal axis, what is happening to occur? Look, a curve. Here it is. Also flattening are like that right is flattening out. At which point trace it to the why access, right? Interesting to the white answers that you wish your graphing calculator. You can see that here is 1.4 positive. 1.4. Right. So the limits as X ghost infinity off those function above here does function Here is approximately 1.4. Right. That is part a graph. So we can now use tables to also see if we can confirm the same thing. Right? Just like we did in the last you tolerate before this one, right? So if you use tables, I mean, you can start from 10. Whatever. Right? Ah, let me start from, uh, tend to the five. I can start concerned from 10. You concerned from, uh, 10 in 10 square in 10 Cube. But I&#x27;m starting from 10 to the five. Mean just what Every c x and dysfunction put in 10 to the five, right, and then put it in a crock leader. And what are you gonna get while you&#x27;re gonna get 1.4433? That is for 10 to 5. For for 33 When you put intend to the six. What are you gonna get? I&#x27;m gonna get 1.44 337 1.44337 Okay, The 1st 2 previous. One was full for 33769 Mrs. Thursday, 75 Right. I put in 10 to 7. What do you get? You get one point 443371.44337 right. So you can see that you&#x27;re getting closer to you. 1.4, right. Getting closer to one point for as we go forward. Right. So if you keep on coming forward can see that this one is approaching 1.4 right at some point is approaching one point for right. So the limits as executions, infinity of the function for this, uh, table, right, it&#x27;s approximately 1 24 now lets you let&#x27;s use theology, break method out. You bring way off, um, off finding it. Right. So you know how to do it. Algae. Brickley. Right. I want to find the limits. X approaches infinity off those function. What is the function? Ah, square root off three X squared by three x squared close. Uh, eight experts 688 X close. Six, then minus screaming of three x squared. Plus three x Then I think plus one. Yeah. So you know how to do it. You just multiply by the congregation, right? I don&#x27;t have a lot of space here, but you multiply by the congregation, both the top and the bottom, right? So when you do that, what you&#x27;re gonna happen? What is gonna happen is that the top one? You know, the top one is gonna be the first terms where if you&#x27;ve been following it, that story, you know exactly what I&#x27;m doing. The first time squared, which is the same. Let&#x27;s just whatever is that a radical under the radical sign? The mind is a second term square, which is also the same as the things on under the radical like that I put in parentheses because it&#x27;s not just 11 thing, and then over. And what is the denominator is, uh, the original thing, right? Remember, mortified this whole thing by this over the same thing. That is what got us to this place, right? And then you try to find the limits as executions Infinity off this this thing eso You know, this negative here is gonna make each and every term here negative, right? So if their own negative, what is happening now? You have three X squared minus three. Expert goes away, does zero right? And then you have ah, eight X minus three X, which is five X. Right. And they have six minus one, which is five. So this has become just becomes five X minus five. Right? And then you have this denominator here, right? And you have the limits. Exit pushes, symphony. So we have to do is deal with the denominator, right? I&#x27;m gonna flap throughout. Uh, X squared right and found around X ray here as well. So when I do that, what is happening? When I do that, I&#x27;m gonna get I&#x27;m going it five X minus five over. This is gonna be x X squared out. I have been a factor out X squared and have three plus eight over X. The six over X squared right under the square feet. Other miners I do the same thing here. X squared out right. Then I get a three plus three over X plus one over X were right. Then when I pull it out What I did, I get absolute value of X. Remember? If that was the last tutorial before this one, please watch it I explained things there. This is gonna be absolute value of X, right, Because when we pull out as X squared from the square, which is gonna be squared off X squared like that out, right, and then this one is the same as absolute value of X. They&#x27;re gonna determine what kind of infinity go into to determine whether to make it positive or negative, right? In fact, let me just go through it in order to ensure that you understand. So what I&#x27;m doing here is I fact it out. This this exquisite out. So that is what is having here, right? So once I do that, then I have three plus eight over X six over X squared A do same here, fans around the X squared, right? I have fact around this one. Did I have Ah, three plus three over X plus one over X squared, Right. This one I have Now, what is the square root off X squared? It issues X right, but you put it in absolute value of X because he did not know exactly what excess. And it&#x27;s gonna be negative and positive. Right? So you put it in absolute value of X, right? And so when you do that, they&#x27;re gonna have here three pills, eight over experts, six x squared. The same vein here becomes after the value of X. And then you have ah, three plus three over X plus one over X squared. And they have one top five X minus five. Right. So once you have this one than what is happening, then you look at the limit. The limit is what is gonna determine what kind of value it is wiser. The limit as X approaches infinity. So excess approaching positive infinity. So we look at the absolute value. The X value is positive. So you can you can safely take away the absolute value would have added any negative, right? So this becomes yes, X because exits positive. So it&#x27;s just X. If X was approaching negative infinity, then it would have been negative. X remember the last tutorial. So this one here becomes three plus eight of X because six over X squared, right, and then, um and then minus also X because it is approaching positive. Infinity. This is three plus three over X plus one over X squared, and it is what becomes that now? The highest power at the denominator is exit apartment one. Right. So I divide each and return by exited power one. So this becomes five X over X minus five X. Right. And then this one becomes, uh, three plus eight over X plus six X squared. Now divide this one over X. Right, And I divide this 12 over X right? I spent about you for X, so I don&#x27;t buy that over excess. Well, and then here becomes Ah, three, three. Right. Those three or x plus one over X squared right now what I have and I have this cancer than this one that I have five minus Final Four X right. And then he becomes, ah, just three plus eight over X plus six over X squared. Then here becomes three plus three over X close one of the X squared. So when you take the limit, as X approaches infinity, this one goes to zero. This goes to zero. This goes to zero. This goes to zero. And this ghost So you have finally is five over. Um Okay, so so, uh, here it&#x27;s supposed to be positive. Here the congregation of this one is positive. This is negative suited. Conjugation is positive. Otherwise would have made a mistake. This is positive. This is positive. At as positive throughout is the congregation of the function itself is negative, right? It&#x27;s minus. But the congregation is plus right Said this pose if it was minus it was never your problem because this, uh, screwed or three plus screaming or three. If it was, no president was minus than the denominator is gonna give you zero, which is giving you bigger problems. Right? So it is. It is. It is positive because of the congregation to congregate off this function here is the same thing. But this time this is gonna be Plus, that is what we we did right. Okay, so then this one becomes five five to squint of three. Right? So when you put this one in the Konica later, it&#x27;s approximately 1.44 34 us. Well, hey, so you can see that the limit asked. X goes to infinity off. The function is also approximately 1.4. Right? How about you? Breaking mutilations and paragraph is 1.4 and impertinent. Table is also one for four, right

Problem

Find the limit. $$\lim _{x \rightarrow-3^{+}} \f…

Problem 12 Easy Difficulty

(a) Use a graph of
$$f(x)=\left(1-\frac{2}{x}\right)^{x}$$
to estimate the value of $\lim _{x \rightarrow \infty} f(x)$ correct to two decimal places.
(b) Use a table of values of $f(x)$ to estimate the limit to four decimal places.

Answer

A. $$\lim _{x \rightarrow \infty}\left(1-\frac{2}{x}\right)^{x} \approx 0.135$$
B. $\lim _{x \rightarrow \infty} f(x) \approx 0.135$

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Essential Calculus Early Transcendentals

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A. $$\lim _{x \rightarrow \infty}\left(1-\frac{2}{x}\right)^{x} \approx 0.135$$B. $\lim _{x \rightarrow \infty} f(x) \approx 0.135$

Essential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Kayleah T.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartEssential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Kayleah T.

Kristen K.

A. $$\lim _{x \rightarrow \infty}\left(1-\frac{2}{x}\right)^{x} \approx 0.135$$
B. $\lim _{x \rightarrow \infty} f(x) \approx 0.135$

James Stewart

Essential Calculus Early Transcendentals