Exercise 14

Recall we say that function $g$ dominates function $f$ provided that $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} g(x) = \infty$, and $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$.

1. Which function dominates the other: $\ln(x)$ or $\sqrt{x}$?

2. Which function dominates the other: $\ln(x)$ or $\sqrt[n]{x}$? ($n$ can be any positive integer)

3. Explain why $e^x$ will dominate any polynomial function.

4. Explain why $x^n$ will dominate $\ln(x)$ for any positive integer $n$.

5. Give any example of two nonlinear functions such that neither dominates the other.