Question

use the limit properties to find the following limit, if it exists. lim(x→∞) (x + 10)/(x^3 + 15) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim(x→∞) (x + 10)/(x^3 + 15)= (simplify your answer.) b. the limit does not exist.

Explanation:

Step1: Divide numerator and denominator by $x^3$.

$\lim_{x
ightarrow\infty}\frac{\frac{x}{x^3}+\frac{10}{x^3}}{\frac{x^3}{x^3}+\frac{15}{x^3}}=\lim_{x
ightarrow\infty}\frac{\frac{1}{x^2}+\frac{10}{x^3}}{1 + \frac{15}{x^3}}$

Step2: Use limit properties.

As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}\frac{1}{x^n}=0$ for $n>0$. So $\lim_{x
ightarrow\infty}\frac{\frac{1}{x^2}+\frac{10}{x^3}}{1 + \frac{15}{x^3}}=\frac{0 + 0}{1+0}=0$

Answer:

A. $\lim_{x
ightarrow\infty}\frac{x + 10}{x^3+15}=0$