Ordering by asymptotic growth rates

- Rank the following functions (refer book for the list) by order of growth; that is, find an arrangement , , , of the functions satisfying , , , . Partition your list into equivalence classes such that functions and are in the same class if and only if .
- Give an example of a single non-negative function such that for all functions in part (a), is neither nor .

**1. Rank of The Functions**
Let us first try to simplify as many functions as we can:

The required order of the functions is as follows:

**2. Example Function**
An example of a single non-negative function such that for all functions in part (a), is neither nor is: