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)(-6x^(2)-27x^(6)))/(1+9x^(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)(root(3)(-6x^(2)-27x^(6)))/(1+9x^(2)) Answer:

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Identify highest power: We are given the limit to evaluate: lim_(x → ∞)(root(3)(-6x^(2)-27x^(6)))/(1+9x^(2)) First, we need to identify the highest power of x in the numerator and the denominator to simplify the expression.Factor out x^6 and x^2: The highest power of x in the numerator inside the cube root is x^6, and in the denominator, it is x^2. To simplify, we will factor out x^6 from the cube root in the numerator and x^2 from the denominator.Rewrite the limit: Factoring out x^6 from the cube root in the numerator gives us: root(3)(x^6(-6/x^4-27)) And factoring out x^2 from the denominator gives us: x^2(1+9)Simplify cube root: Now we rewrite the limit as: lim_(x → ∞)(root(3)(x^6(-6/x^4-27)))/(x^2(1+9))Cancel out x^2 terms: We can simplify the cube root of x^6 as x^2 because (x^2)^3 = x^6. The limit now becomes: lim_(x → ∞)(x^2*root(3)(-6/x^4-27))/(x^2(1+9))Simplify expression: We can cancel out the x^2 terms in the numerator and the denominator: lim_(x → ∞)(root(3)(-6/x^4-27))/(1+9)Simplify cube root: As x approaches infinity, the term -6/x^4 approaches 0. So we can simplify the expression inside the cube root to just -27: lim_(x → ∞)(root(3)(-27))/(1+9)Final simplification: The cube root of -27 is -3, and 1+9 is 10. So the limit simplifies to: lim_(x → ∞)(-3)/10Evaluate the limit: Since -3/10 is a constant, the limit as x approaches infinity is simply: -3/10

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-6 x^{2}-27 x^{6}}}{1+9 x^{2}}$ $\newline$ 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). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-6 x^{2}-27 x^{6}}}{1+9 x^{2}}$ $\newline$ Answer: