Question

determine the following limit.
lim_{x
ightarrowinfty}\frac{30x^{3}+3x^{2}-3x}{35x^{3}+x^{2}+4x + 1}
select the correct choice below and, if necessary, fill in the answer box to complete your choice
\text{a. } lim_{x
ightarrowinfty}\frac{30x^{3}+3x^{2}-3x}{35x^{3}+x^{2}+4x + 1}= (simplify your answer.)
\text{b. the limit as }x\text{ approaches }infty\text{ does not exist and is neither }infty\text{ nor }-infty.

Explanation:

Step1: Divide by highest - power term

Divide numerator and denominator by $x^{3}$: $\lim_{x
ightarrow\infty}\frac{30 + \frac{3}{x}-\frac{3}{x^{2}}}{35+\frac{1}{x}+\frac{4}{x^{2}}+\frac{1}{x^{3}}}$

Step2: Evaluate limit

As $x
ightarrow\infty$, $\frac{1}{x},\frac{1}{x^{2}},\frac{1}{x^{3}}
ightarrow0$. So we get $\frac{30 + 0 - 0}{35+0 + 0+0}=\frac{30}{35}=\frac{6}{7}$

Answer:

A. $\frac{6}{7}$