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This is problem number 14 of the stewart calculus 8th edition, section 2 .6.
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Evaluate the limit and justify each step by indicating the appropriate properties of limits.
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The limit is x approaches infinity of the square root of the following quantity, 9x cubed plus 8x minus 4, divided by the quantity 3 minus 5x plus x cubed.
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So we proceed with our calculations by applying properties of limits, as well as making some other simplifications.
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Our first application will be to divide each term by x cubed.
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So 9x cubed divided by x cubed is just 9.
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8x divided by x cubed is 8 over x squared.
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And then negative 4 divided by x cubed and negative 4 over x cubed.
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In the denominator, we have 3 over x cubed minus 5x divided by x cubed.
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Well that's 5 negative 5 x squared and then xx squared is of course just one.
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Our next step is to use properties of limits...