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(a) Use a graph of $$ f(x) = \sqrt{3x^2 + 8x + 6} - \sqrt{3x^2 + 3x + 1} $$ to estimate the value of

$ \displaystyle \lim_{x \to \infty} f(x) $ to one decimal place.

(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

(c) Find the exact value of the limit.

$$

1.4433

$$

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

this problem or forty six of the Stuart Calculus. Eighth Addition. Section two point six party use A graph of FX is he called, too. The square root of the quantity three X squared per se eight x for six minus the square root of the quantity. Three. X squared plus three x plus one to estimate the value of the limiters expertise. Infinity of half to one desolate place. So he's in a graphing calculator. We and put the function into this calculator, and we plot the function and take a look at what the function seems to level off at what the horizontal asked It will be. And here we get a valuable around one point four four six. So to answer the first part two one decimal place, we estimate this limit to be approximately one point for in party. We're going to improve our estimate accuracy using a table values of F or an estimate the limits of Ford to swap places. So here we have a table valleys with numbers that are increasing towards infinity. And then we have the function here and put to each of the cells, and we see that the function Ah, purchase this value much more accurate than one point four. It's exactly protein one point four four three four. So we improve our estimate and party to one point four four three four. Our party, I estimate one point for for three. Four. Okay, in part C, we're going to find the exactly the limit on the way we do. That is by yeah, working with this, taking the limit as experts affinity and our first step will be to multiplying that tap in the bottom by the controversy. It's so three x squared looks eight x six Mine plus the other square returned three x squared plus three x plus one. And that's something we're gonna be able to buy into the top hands everybody First one, of course to X squared, plus a dicks six but square root of to be X squared plus three x plus One came. So when we don't want to play this out Ah, in cans for a minute. Experts Infinity, Um, here t it's going to foil based on difference of square. So we end up getting just the first of the last terms. Ah, and it's just the what's inside the first square root three x squared the Pacific's six without the square minus the other quantity without the square three X squared plus three x plus one and the denominator remains the same. The square root of each of these terms three. X squared plus eight x six plus square root of re X squared of three x plus one. Okay, At this point, we didn't find that three X squared minus three X squared equals zero other simplifications that can make I have to do with the ah other like terms in the numerator we have eight x minus three x That's five x on DH. Then we also have a six minus one, which is fine. So in the numerator we have Dr Nix plus five end of the nominator we have thie. He's radical still three x squared. Let's say next six. Both Scarlett of doing X squared three X plus one. Now our next step is to divide by X to the generator and the denominator came. So if we do that, you know I'm a writer. Will be paddocks erects. It's fine. Oh, Iraq's and the denominators. Since we have radicals, square root skirts we divided by X, which is the same as divided by the square root of X squared teach term. So if we incorporate that, incorporate that into a tragical we'LL get three x squared over X squared is three eight x r eight X squared or ethics or X squared is a durex and six or X squared as the last term. Russ three. Expert of a three over X squared is three thanks over X squared history over X and then one over X squared as a last year and X approaches infinity. Each of these fractional rallies that are some number over X or some never over X squared approach zero reigned as X goes towards infinity is fractions are negligible and we can the valley We ought to live it now as simplified to five plus he around in a raider and we have scored a three last the square to three in the denominator. So what this equals is five over to attempt the square root of three which, if we use a calculator, is approximately equal to one waiting for or, uh, three, four and there. But in the form that it is, it is the exact huh? I have over two times the square root of three. That is exact value for part, see?