5. Dominance

Hierarchy of Dominance

In order to apply the method of the dominant term we will need to know the relative growth rates of certain elementary functions. This leads to a hierarchy of dominance. The following list is not exhaustive (in fact it cannot be), but hopefully the list will provide you with the tools to place any term you may find in the right spot on the list. In the list below the growth rates of the functions increases as you go from left to right. This list is based on the case where $x\to \infty$ although you should be able to construct a similar list for $x\to -\infty$ based on it.

1. $B$
2. $\ln (x)$
3. $\ln (x{)}^2$
4. $\sqrt{x}$
5. $x$
6. $x\ln (x)$
7. $x^2$
8. $x^3$
9. $e^x$
10. $x^x$
11. $(e^x{)}^2$

Here the letter $B$ refers to bounded functions that do not in fact go to $\infty$ as $x\to \infty$. Examples of these are $\cos (x)$, $\sin (x)$, $e^{-x}$, $5$, $\frac{1}{x}$, etc.