Let $\displaystyle m$ and $\displaystyle n$ be positive integers, c a positive real number. The following functions all tend to $\displaystyle \infty$ as $\displaystyle x \to \infty$ and they are arranged in order of increasing growth rate:

$\displaystyle \ln(x), \! x^{1/m}, \! x^n, \! e^{cx}, \! e^{x^{n}}$

(a) Where in the list would you place $\displaystyle x^x$ and $\displaystyle \frac{x}{\ln(x)}?$

(b) Find a function which grows faster than any on the list.

(c) Find a function which grows more slowly than any on the list.