Problem 9 Medium Difficulty

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}} $

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04:18

Daniel J.

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to evaluate this limit, We first apply limit of a route. That is if we have limit as X approaches he ah the ends route of F of X, this is equal to the answer route of the limit of F of X as X approaches E. And so from here we can rewrite this as the square it of the limit as X approaches to of two X squared plus one over three X -2. Next you want to apply limit of a Kocian? That is if we have limit as X approaches a. Of the cohesion of F X and G fx this is equal to the caution of their limits. That's limit as X approaches a of F of X over limit as X approaches A. Of G of X. And so from here we have the square it of we have limits as X approaches to of two, X squared plus one fish all over the limit as X approaches to of three X -2. And the next we want to apply limit of a sum and difference that is. If we have the limit of the sum or difference of F and G as X approaches E then we have the sum or difference of their limits. That limits as X approaches a of f of X plus or minus a limit as X approaches A of G of x and so we have the square it of the limit as X approaches to of two x squared plus The limit as X approaches to have one. This all over the limit as X approaches to of three x minus the limit as X approaches to of two. Next we want to apply the limit for a constant multiple. That is if we have the limit of see of F of X as X approaches a this is equal to C times the limit as X approaches a off F of X and so and here we have the square it of limit as X approaches to of X squared this times two plus we have limit as X approaches to of one all over we have three times the limit as X approaches to of x minus Limit as X Approaches two of 2. And then lastly, you want to apply the limit of a power that is if we have the limit of F of x rays to the ends power as X approaches a This is equal to the limit of F of X as X approaches a this whole thing raised to the ends power. And so from here we have the square it of two times the limit as X approaches to of X and then this whole thing squared plus we have the limit as X approaches to of one over three times the limit as x approaches to of x minus limit as X approaches to to and know that the limit of the constant is equal to that constant itself. And so this is equal to the square it of we have two times 2 squared plus we have one over three times 2 -2 two, Which is equal to the square root of nine over four, or that's just 3/2. And so this is the value of the limit.