Question

find $lim_{x
ightarrowinfty}\frac{5x^{3}+2x^{2}-7}{x^{4}+3x}$. choose 1 answer: a 0 b 5 c $\frac{3}{4}$ d the limit is unbounded

Explanation:

Step1: Divide by highest - power of x in denominator

Divide both the numerator and denominator by $x^{4}$, since the highest - power of $x$ in the denominator is $x^{4}$.
We get $\lim_{x
ightarrow\infty}\frac{\frac{5x^{3}}{x^{4}}+\frac{2x^{2}}{x^{4}}-\frac{7}{x^{4}}}{\frac{x^{4}}{x^{4}}+\frac{3x}{x^{4}}}=\lim_{x
ightarrow\infty}\frac{\frac{5}{x}+\frac{2}{x^{2}}-\frac{7}{x^{4}}}{1 + \frac{3}{x^{3}}}$.

Step2: Use limit rules

We know that $\lim_{x
ightarrow\infty}\frac{c}{x^{n}} = 0$ for any constant $c$ and positive integer $n$.
So, $\lim_{x
ightarrow\infty}\frac{\frac{5}{x}+\frac{2}{x^{2}}-\frac{7}{x^{4}}}{1+\frac{3}{x^{3}}}=\frac{0 + 0-0}{1 + 0}$.

Answer:

A. 0