find the limit, if it exists\nlim(x→∞) (-4x³ + 4x)/(9x² - 8x - 5)\nselect the correct answer below and, if…

Explanation:

Step1: Divide by highest - power of x in denominator

Divide both the numerator and denominator by $x^{2}$ since the highest - power of $x$ in the denominator is $x^{2}$. We get $\lim_{x\rightarrow\infty}\frac{-4x^{3}/x^{2}+4x/x^{2}}{9x^{2}/x^{2}-8x/x^{2}-5/x^{2}}=\lim_{x\rightarrow\infty}\frac{-4x + \frac{4}{x}}{9-\frac{8}{x}-\frac{5}{x^{2}}}$.

Step2: Evaluate the limit of each term

As $x\rightarrow\infty$, $\lim_{x\rightarrow\infty}\frac{4}{x}=0$, $\lim_{x\rightarrow\infty}\frac{8}{x}=0$, and $\lim_{x\rightarrow\infty}\frac{5}{x^{2}} = 0$. So, $\lim_{x\rightarrow\infty}\frac{-4x+\frac{4}{x}}{9 - \frac{8}{x}-\frac{5}{x^{2}}}=\lim_{x\rightarrow\infty}\frac{-4x+0}{9 - 0 - 0}=\lim_{x\rightarrow\infty}\frac{-4x}{9}=-\infty$.

Answer:

B. The limit does not exist and is neither $-\infty$ nor $+\infty$.