Ordering by asymptotic growth rates

1. Rank the following functions (refer book for the list) by order of growth; that is, find an arrangement $g_1$, $g_2$, $\cdots$ , $g_{30}$ of the functions satisfying $g_1 = \Omega(g_2)$, $g_2 = \Omega(g_3)$, $\cdots$ , $g_{29} = \Omega(g_{30})$. Partition your list into equivalence classes such that functions $f(n)$ and $g(n)$ are in the same class if and only if $f(n) = \Theta(g(n))$.
2. Give an example of a single non-negative function $f(n)$ such that for all functions $g_i(n)$ in part (a), $f(n)$ is neither $O(g_i(n))$ nor $\Omega(g_i(n))$.

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 $f(n)$ such that for all functions $g_i(n)$ in part (a), $f(n)$ is neither $O(g_i(n))$ nor $\Omega(g_i(n))$ is:

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