Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-27 x^{7}-x^{12}}}{8 x^{2}+x+8 x^{4}}$

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Multiplication with rational exponents

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Q. Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).

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\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-27 x^{7}-x^{12}}}{8 x^{2}+x+8 x^{4}}

Factor Out Highest Powers: We are given the limit: $\newline$ $\lim_{x \to \infty}\left(\frac{\sqrt[3]{-27x^{7}-x^{12}}}{8x^{2}+x+8x^{4}}\right)$ $\newline$ To simplify this limit, we will first factor out the highest power of $x$ in the numerator and denominator.
Simplify Inside Cube Root: In the numerator, the highest power of $x$ is $x^{12}$ , and in the denominator, it is $x^4$ . We will factor these out from the root and the polynomial respectively. $\newline$ Numerator: $\sqrt[3]{x^{12} * (-\frac{27}{x^5} - 1)}$ $\newline$ Denominator: $x^4 * (\frac{8}{x^2} + \frac{1}{x^3} + 8)$
Approach Infinity: Now we simplify the expression inside the cube root in the numerator and the terms in the denominator. $\newline$ Numerator: $x^4 \cdot \sqrt[3]{-\frac{27}{x^5} - 1}$ $\newline$ Denominator: $x^4 \cdot \left(\frac{8}{x^2} + \frac{1}{x^3} + 8\right)$ $\newline$ As $x$ approaches infinity, the terms with negative powers of $x$ will approach zero.
Simplify Numerator and Denominator: After the terms with negative powers of $x$ approach zero, we have: $\newline$ Numerator: $x^4 \cdot \sqrt[3]{-1}$ $\newline$ Denominator: $x^4 \cdot 8$
Cancel Out $x^4$ Terms: The cube root of $-1$ is $-1$ , so the numerator simplifies to: $\newline$ Numerator: $x^4 \cdot (-1)$ $\newline$ Denominator: $x^4 \cdot 8$
Final Simplified Form: Now we can cancel out the $x^4$ terms in the numerator and denominator: $\newline$ Numerator: $-1$ $\newline$ Denominator: $8$
Final Simplified Form: Now we can cancel out the $x^4$ terms in the numerator and denominator: $\newline$ Numerator: $-1$ $\newline$ Denominator: $8$ The final simplified form of the limit is: $\newline$ $\lim_{x \to \infty}(-\frac{1}{8})$ $\newline$ Since this is a constant, the limit as $x$ approaches infinity is simply the constant value.