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Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).

$ \displaystyle \lim_{x \to 2}\sqrt{\frac{2x^2 + 1}{3x - 2}} $

$\frac{3}{2}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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this problem. Number nine of the Stuart Calculus eighth Edition section two point three. Evaluate the limit and justify each them by indicating the appropriate limit loss. The limit is expertise to of the square root of the quantity to X squared, plus one divided by the quantity. Three X minus two. Now the first thing to address here is the radical, and we use the limitless associated with radicals that tells us that the element of a function with a radical our ah function under a radical is the same. It's that same limit or is the same as the radical in this case, two square root of limit as experts is too, of this quotient to X squared plus one directed by three x minus two. And here we have used a limit law associated with radicals. We Now here's the limit losses here with division. We use the quotient limitless. And under this radical, we're able to split this limit of a question or a ratio into one limit. You had to buy another one in this case generator. The limitless expert. Just two of the quantity to X squared, plus one divided right. The LTD's expressions too, with the quantity three X minus two. Finally, we will be using the sum indifference limit laws to separate these limits and simple for them even further. Here we have the limit of two x squared or using the concept it constant multiple law. You get two times the limit of X squared, and then here the next part is plus the limited one. And in the denominator we have two terms. We will have two terms to limits. This is the limit of three X, but again using the constant multiple on its three times limited X minus, their remaining limit of the function to the constant function, too. Now we can use what the limit, what each of these is. Um, it's our representative. Lim has expertise to. And if we begin in the numerator under that radical, two times the limit as expressions, too, of X squared, so just be too square class. The limit is expressions to one. It is equal to one all divided by three times. The limit is X approaches to X so three ten with two minus the ltd's expertise to of to which is too the next temple. Be too simplify. And you know, they wait. This these numbers two times two squared is eight. This one. You ready, Brian? Three times six, minus two. This will be the squared of nine over for which gives us a final answer of three over too.

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