Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 29

$$1$$

\begin{align*} \lim _{x \rightarrow-\infty} \frac{\sqrt[3]{x}-\sqrt[5]{x}}{\sqrt[3]{x}+\sqrt[5]{x}} : \frac{\sqrt[3]{x}}{\sqrt[3]{x}}&=\lim _{x \rightarrow-\infty} \frac{1-\frac{x^{1 / 5}}{x^{1 / 3}}}{1+\frac{x^{1 / 5}}{x^{1 / 3}}}\\ &=\lim _{x \rightarrow-\infty} \frac{1-\frac{1}{x^{2 / 15}}}{1+\frac{1}{x^{2 / 15}}}\\ &= \frac{\lim _{x \rightarrow-\infty}(1)-\lim _{x \rightarrow-\infty}\frac{1}{x^{2 / 15}}}{\lim _{x \rightarrow-\infty}(1)+\lim _{x \rightarrow-\infty}\frac{1}{x^{2 / 15}}}\\ &=\frac{1-0}{1+0}\\ &=1 \end{align*}