$[$Asymptotics$,25$ points$]$

Sort the following $20$ functions from asymptotically smallest to asymptotically largest, indi$-$

cating ties if there are any. You must provide proofs that justify your answers to receive full

credit! Nll logarithms are base $2$ unless otherwise stated.

$n^{4\mathrm{.}5}-(n-1{)}^{4\mathrm{.}5},nlglgn,n\sqrt[\phantom{2}]{\frac{n}{2}},n^{2n},lg(l{g}^{*}n)$

$n^{\frac{lglgn}{lgn}},l{g}^{(9001)}n,l{g}^{*}{2}^n,lg(n!),(lgn{)}^{lgn}$

${\displaystyle \underset{i=2}{\overset{n}{}}}\frac{2}{2^2-1},e^{2n},n^{lglgn},n^{1337},l{g}^{*}(\frac{n}{2})$

$(1+\frac{1}{787898}{)}^{787898n},n!,,2^{l{g}^{*}n},(lgn{)}^{l{g}^{*}n}$

To simplify notation, write $f(n)g(n)$ to mean $f(n)=o(g(n))$ and $f(n)-=g(n)$ to mean

$f(n)=(g(n)).$ For example, the functions $n^2,n,(\frac{n}{2}),n^3$ could be sorted cither as $n{n}^2-=$

$(\frac{n}{2})n^3$ or as $n(\frac{n}{2})-=n^2{n}^3.$