divide numerator and denominator by the highest power of x\n lim_{x\rightarrow-infty}left(\frac{1…

Explanation:

Step1: Identify highest power of x

The highest power of $x$ in the denominator $x^{2}+8x$ and numerator $1 - x^{3}$ is $x^{3}$.

Step2: Divide numerator and denominator by $x^{3}$

We have $\lim_{x\rightarrow-\infty}\left(\frac{\frac{1}{x^{3}}-\frac{x^{3}}{x^{3}}}{\frac{x^{2}}{x^{3}}+\frac{8x}{x^{3}}}\right)^{7}=\lim_{x\rightarrow-\infty}\left(\frac{\frac{1}{x^{3}} - 1}{\frac{1}{x}+\frac{8}{x^{2}}}\right)^{7}$.

Step3: Evaluate limits of individual fractions

As $x\rightarrow-\infty$, $\lim_{x\rightarrow-\infty}\frac{1}{x^{3}} = 0$, $\lim_{x\rightarrow-\infty}\frac{1}{x}=0$, $\lim_{x\rightarrow-\infty}\frac{8}{x^{2}} = 0$. So the expression inside the parentheses approaches $\frac{0 - 1}{0+0}=-\infty$.

Step4: Raise to the 7th - power

Since $(-\infty)^{7}=-\infty$, the value of $\lim_{x\rightarrow-\infty}\left(\frac{1 - x^{3}}{x^{2}+8x}\right)^{7}=-\infty$.

Answer:

$-\infty$