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Problem 14 Easy Difficulty

Evaluate the limit and justify each step by indicating the appropriate properties of limits.

$ \displaystyle \lim_{x \to \infty} \sqrt{\dfrac{9x^3 + 8x - 4}{3 - 5x + x^3}} $

Answer

3

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Video Transcript

this problem over fourteen of the story calculus eighth edition Section two point six in Hollywood Limit interested Aren't you step by indicating the appropriate properties of living? The limit is ex emerges infinity of the square roots of the following quantity and execute Pot's eight explains for divided by the quantity three minus five. Expose. Excuse. So we proceed with our calculations by applying properties limits as well as making some other simplification. Our first occassion. It will be, too. Divide each term. Right? Executed The nanny Cubed, Divided Excuses. Just nine. Eight Extra out of excuses. Needlework Square Now and then. Negative Florida Out of executed Think forever you the denominator. We have three over X cubed. It's five x, The Betamax usual. That's right. Thinking about it will record squared and the next you know how to execute is, of course, just one. Yeah, Our next step is to use properties of limits. We can remove this or move this square. Retain, which is a root, are an exponent of equal tohave to the outside. This limit so will be the same images raised that one have power and then within the traction will be able to split each limit according to the property of Limit. Where Theoden shin of terms instead of a limit, can translates to the summer those limits individually, so limit his expertise. Infinity of nine Let's limit is a capricious affinity of eight or X squared, minus the limit. What experts? Infinity of four over X cubed. That is just the numerator. The denominator is eliminates explosions. Infinity of the quantity. Three. Over Execute finest Limited's explosion Infinity of the quantity over X squared. I'm classy. Women is expressive in the quantity one and as, uh, we can recall Well, um, it is expression infinity of any quantity. Uh, it looks like this one A rex to the arm where are is a rational number great zero, or is limited quiz era. So each of these terms here has ah denominator for a constant in the numerator and the denominator of X or X squared or X, cubed all these limits of expression and go to zero. So each of these women go zero vanish. We are left with limited express infinity, uh, for a nine, which is a constant value. That's a sequel tonight. Plus hear of us zero is just see Rome. And then in the denominator, we have zero minus zero. Plus the limited experience. Unity of their cocks value one. And that woman is equal to one. So all we're left with is my number one, which is nine and a final answer being square root over thing, which is three. That is our final answer for limit. All right, He really really meant Italy utilizing the appropriate property. The limits for addition. Ah, division subtraction and powers.