?Find the limit or show that it does not exist.\(\lim _{r \rightarrow \infty}

Step 1 of 2

Consider the given function as

\(\lim _{r \rightarrow \infty} \frac{r-r^{3}}{2-r^{2}+3 r^{3}}\)

Divide both the numerator and the denominator by the highest power of r of the denominator.

\(\begin{aligned} \lim _{r \rightarrow \infty} \frac{r-r^{3}}{2-r^{2}+3 r^{3}} & =\lim _{r \rightarrow \infty} \frac{\frac{r}{r^{3}}-\frac{r^{3}}{r^{3}}}{\frac{2}{r^{3}}-\frac{r^{2}}{r^{3}}+3 r^{3}} \\ & =\lim _{r \rightarrow \infty} \frac{\frac{1}{r^{2}}-1}{\frac{2}{r^{3}}-\frac{r^{2}}{r^{3}}+\frac{3 r^{3}}{r^{3}}} \end{aligned}\)