Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). lim_(x rarr oo)(root(3)(-8x^(14)+16 x+x^(8)))/(9x^(4)+9x) 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)(root(3)(-8x^(14)+16 x+x^(8)))/(9x^(4)+9x) Answer:

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Identify highest power: We are given the limit expression: lim_(x -> ∞)(root(3)(-8x^(14) + 16x + x^(8)))/(9x^(4) + 9x) To simplify this 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 cube root is x^(14), and the highest power of x in the denominator is x^(4). To simplify the limit, we will divide both the numerator and the denominator by x^(4), the highest power in the denominator.Factor out x^4: We rewrite the limit expression by factoring out x^(4) from the denominator and x^(12) (which is (x^(4))^3) from the numerator inside the cube root: lim_(x -> ∞)(root(3)(x^(12)(-8x^(2) + 16/x^(10) + 1/x^(4))))/(x^(4)(9 + 9/x^(3)))Simplify inside cube root: Now we simplify the expression inside the cube root by dividing each term by x^(12): lim_(x -> ∞)(root(3)(-8 + 16/x^(22) + 1/x^(16)))/(9 + 9/x^(3))Ignore terms approaching zero: As x approaches infinity, the terms 16/x^(22), 1/x^(16), and 9/x^(3) will approach zero. Therefore, we can ignore these terms for the limit calculation: lim_(x -> ∞)(root(3)(-8))/(9)Calculate cube root: The cube root of -8 is -2, so the limit simplifies to: lim_(x -> ∞)(-2)/9Simplify constant limit: Since -2/9 is a constant, the limit as x approaches infinity is simply: -2/9

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-8 x^{14}+16 x+x^{8}}}{9 x^{4}+9 x}$ $\newline$ 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-8 x^{14}+16 x+x^{8}}}{9 x^{4}+9 x}$ $\newline$ Answer: