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)(-19x^(3)-27x^(6)))/(5x+9x^(2)+9) 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)(-19x^(3)-27x^(6)))/(5x+9x^(2)+9) Answer:

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Factor out highest power: We are given the limit expression lim_(x -> ∞)(root(3)(-19x^(3)-27x^(6)))/(5x+9x^(2)+9). To simplify this, we will first factor out the highest power of x in the numerator and denominator.Rewrite with factored terms: In the numerator, the highest power of x inside the cube root is x^6. We factor out x^6 from each term inside the cube root. root(3)(-19x^(3)-27x^(6)) = root(3)(x^6(-19/x^3-27))Simplify cube root: In the denominator, the highest power of x is x^2. We factor out x^2 from each term. 5x+9x^(2)+9 = x^2(5/x+9+9/x^2)Divide by x^2: Now we rewrite the limit expression with the factored terms. lim_(x -> ∞)(root(3)(x^6(-19/x^3-27)))/(x^2(5/x+9+9/x^2))Approaching infinity: We can simplify the cube root of x^6 as x^2 because (x^2)^3 = x^6. root(3)(x^6(-19/x^3-27)) = x^2 * root(3)(-19/x^3-27)Find limit: Now we divide both the numerator and the denominator by x^2. lim_(x -> ∞)(x^2 * root(3)(-19/x^3-27))/(x^2(5/x+9+9/x^2)) = lim_(x -> ∞)(root(3)(-19/x^3-27))/(5/x+9+9/x^2)Calculate cube root: As x approaches infinity, the terms with x in the denominator approach 0. lim_(x -> ∞)(root(3)(-19/x^3-27)) = root(3)(-27) lim_(x -> ∞)(5/x+9+9/x^2) = 9Divide to get limit: Now we can find the limit of the simplified expression. lim_(x -> ∞)(root(3)(-19/x^3-27))/(5/x+9+9/x^2) = root(3)(-27)/9The cube root of -27 is -3. root(3)(-27) = -3Finally, we divide -3 by 9 to get the limit. -3/9 = -1/3

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-19 x^{3}-27 x^{6}}}{5 x+9 x^{2}+9}$ $\newline$ 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-19 x^{3}-27 x^{6}}}{5 x+9 x^{2}+9}$ $\newline$ Answer: