Chapter 1 - Limits and Continuity - 1.4 Limits (Discussed More Rigorously) - Exercises Set 1.4 - Page 88: 40

$(\text{a})$ $N = \sqrt[3]{10}$. $(\text{b})$ $N = \sqrt[3]{100}$. $(\text{c})$ $N = \sqrt[3]{1000} = 10$.

The condition that $|1/x^3-0| < \epsilon$ is can be expressed as $1/x^3 < \epsilon$ for $x$ positive, which is equivalent to $x > \sqrt[3]{1/\epsilon}$. Thus $N = \sqrt[3]{1/\epsilon}$.