Question

use the limit properties to find the following limit, if it exists.
lim_{x
ightarrowinfty}\frac{x^{2}+13}{x^{3}+9}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. lim_{x
ightarrowinfty}\frac{x^{2}+13}{x^{3}+9}= (simplify your answer.)
b. the limit does not exist.

Explanation:

Step1: Divide by highest - power

Divide numerator and denominator by $x^{3}$: $\lim_{x
ightarrow\infty}\frac{\frac{x^{2}}{x^{3}}+\frac{13}{x^{3}}}{\frac{x^{3}}{x^{3}}+\frac{9}{x^{3}}}=\lim_{x
ightarrow\infty}\frac{\frac{1}{x}+\frac{13}{x^{3}}}{1 + \frac{9}{x^{3}}}$

Step2: Apply limit rules

As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}\frac{1}{x}=0$ and $\lim_{x
ightarrow\infty}\frac{13}{x^{3}} = 0$, $\lim_{x
ightarrow\infty}\frac{9}{x^{3}}=0$. So $\lim_{x
ightarrow\infty}\frac{\frac{1}{x}+\frac{13}{x^{3}}}{1+\frac{9}{x^{3}}}=\frac{0 + 0}{1+0}$

Answer:

A. $0$