Question

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dn

$$lim _ { r ightarrow infty } \frac { r - r ^ { 3 } } { 4 - r ^ { 2 } + 8 r ^ { 3 } }$$

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Explanation:

Step1: Divide numerator and denominator by \(r^{3}\)

$$\lim_{r ightarrow\infty}\frac{\frac{r}{r^{3}}-\frac{r^{3}}{r^{3}}}{\frac{4}{r^{3}}-\frac{r^{2}}{r^{3}}+\frac{8r^{3}}{r^{3}}}=\lim_{r ightarrow\infty}\frac{\frac{1}{r^{2}} - 1}{\frac{4}{r^{3}}-\frac{1}{r}+8}$$

Step2: Apply the limit

As \(r
ightarrow\infty\), \(\lim_{r
ightarrow\infty}\frac{1}{r^{2}} = 0\), \(\lim_{r
ightarrow\infty}\frac{4}{r^{3}}=0\), \(\lim_{r
ightarrow\infty}\frac{1}{r}=0\)
$$\frac{0 - 1}{0-0 + 8}=-\frac{1}{8}$$

Answer:

\(-\frac{1}{8}\)