Question

find the limit, if it exists.
lim(x→∞) (-6x^4 + 7x)/(8x^3 - 8x - 9)
select the correct answer below and, if necessary, fill in the answer box to complete your choice.
a. lim(x→∞) (-6x^4 + 7x)/(8x^3 - 8x - 9) =
(simplify your answer.)
b. the limit does not exist and is neither -∞ nor +∞.

Explanation:

Step1: Divide numerator and denominator by highest - power of x in denominator

Divide $\frac{-6x^{4}+7x}{8x^{3}-8x - 9}$ by $x^{3}$. We get $\lim_{x
ightarrow\infty}\frac{\frac{-6x^{4}}{x^{3}}+\frac{7x}{x^{3}}}{\frac{8x^{3}}{x^{3}}-\frac{8x}{x^{3}}-\frac{9}{x^{3}}}=\lim_{x
ightarrow\infty}\frac{-6x+\frac{7}{x^{2}}}{8-\frac{8}{x^{2}}-\frac{9}{x^{3}}}$.

Step2: Evaluate the limit of each term

As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}\frac{7}{x^{2}} = 0$, $\lim_{x
ightarrow\infty}\frac{8}{x^{2}}=0$, $\lim_{x
ightarrow\infty}\frac{9}{x^{3}} = 0$, and $\lim_{x
ightarrow\infty}(-6x)=-\infty$. So, $\lim_{x
ightarrow\infty}\frac{-6x+\frac{7}{x^{2}}}{8-\frac{8}{x^{2}}-\frac{9}{x^{3}}}=-\infty$.

Answer:

A. $\lim_{x
ightarrow\infty}\frac{-6x^{4}+7x}{8x^{3}-8x - 9}=-\infty$