Question

give a limit expression that describes the left end behavior of the function.
f(x) = \frac{9 + 5x + 5x^{3}}{x^{3}}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \lim_{x\to -\infty} \frac{9 + 5x + 5x^{3}}{x^{3}} =

b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Simplify the function

Divide each term in the numerator by $x^{3}$: $\frac{9 + 5x+5x^{3}}{x^{3}}=\frac{9}{x^{3}}+\frac{5x}{x^{3}}+\frac{5x^{3}}{x^{3}}=\frac{9}{x^{3}}+\frac{5}{x^{2}} + 5$.

Step2: Find the limit as $x\to-\infty$

We know that $\lim_{x\to-\infty}\frac{9}{x^{3}}=0$ (since the degree of the denominator is positive) and $\lim_{x\to-\infty}\frac{5}{x^{2}} = 0$. Then $\lim_{x\to-\infty}(\frac{9}{x^{3}}+\frac{5}{x^{2}}+5)=\lim_{x\to-\infty}\frac{9}{x^{3}}+\lim_{x\to-\infty}\frac{5}{x^{2}}+\lim_{x\to-\infty}5$.

Step3: Calculate the final - limit

$0 + 0+5 = 5$.

Answer:

A. $\lim_{x\to-\infty}\frac{9 + 5x+5x^{3}}{x^{3}}=5$