Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). lim_(x rarr oo)(sqrt(64x^(9)+39 x+39x^(8)))/(9x+2x^(4)+4x^(2)) Answer:

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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).  lim_(x rarr oo)(sqrt(64x^(9)+39 x+39x^(8)))/(9x+2x^(4)+4x^(2)) Answer:

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Identify highest power: We are given the limit: lim_(x -> ∞)(sqrt(64x^(9)+39x+39x^(8)))/(9x+2x^(4)+4x^(2)) To simplify the limit, we will first identify the highest power of x in the numerator and the denominator.Divide by x^4: The highest power of x in the numerator inside the square root is x^9, and the highest power of x in the denominator is x^4. To simplify the expression, we will divide both the numerator and the denominator by x^(4).Rewrite the limit: We rewrite the limit as: lim_(x -> ∞)(sqrt((64x^(9)/x^(4))+(39x/x^(4))+(39x^(8)/x^(4))))/((9x/x^(4))+(2x^(4)/x^(4))+(4x^(2)/x^(4)))Simplify terms: Simplify the terms inside the square root and the terms in the denominator: lim_(x -> ∞)(sqrt(64x^(5)+39/x^(3)+39x^(4)))/(9/x^(3)+2+4/x^(2))Ignore negative powers: As x approaches infinity, the terms with negative powers of x will approach zero. Therefore, we can ignore 39/x^(3) in the numerator and 9/x^(3) and 4/x^(2) in the denominator. lim_(x -> ∞)(sqrt(64x^(5)+39x^(4)))/2Take square root: Now, we can take the square root of the leading term in the numerator, which is 64x^(5), since the other term, 39x^(4), becomes negligible as x approaches infinity. lim_(x -> ∞)(sqrt(64x^(5)))/2Simplify expression: The square root of 64x^(5) is 8x^(5/2). So the limit simplifies to: lim_(x -> ∞)(8x^(5/2))/2Divide by 2: Simplify the expression by dividing 8x^(5/2) by 2: lim_(x -> ∞)(4x^(5/2))Limit is infinite: Since x^(5/2) grows without bound as x approaches infinity, the limit is infinite. Therefore, the limit does not exist (DNE).

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{64 x^{9}+39 x+39 x^{8}}}{9 x+2 x^{4}+4 x^{2}}$ $\newline$ Answer:

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