$3]$ Rank the following functions by order of growth; that is$,$ find an arrangement

$g_1,g_2,...,g_{17}$

of the functions below satisfying

$g_1$inO$(g_2),g_2$inO$(g_3),...,g_9$inO$(g_{17}).$

Partition your list into equivalence classes such that $f(n)$ and $g(n)$ are in the same class if and only if $f(n)in(g(n)).$ Robust mathematical proofs are preferred, but some credit might be obtained for correct answers obtained through "hand waving" arguments $($the wilder the waving, the lower the grade$).($And note that $"lg{n}^n$ is $"lo{g}_2{n}^n.)$

It may help to observe that if $f(n)ino(g(n)),$ then $f(n)!in(g(n)).$

$($i$)\sqrt[\phantom{2}]{n}$

$($ii$)n^4$

$($iii$){16}^{gn}$

$($iv$)lg((2n)!)$

$($v$)n$

$($vi$)n!$

$($vii$)2^{lgn}{2}^{n-1}$

$($viii$)lgn$

$($ix$)lg(lgn)$

$($x$)\sqrt[\phantom{2}]{2lgn}$

$($xi$)(n+1)!$

$($xii$)\sqrt[3]{2^{-lgn}}$

$($xiii$)300n+2\mathrm{.}99793{10}^8$

$($xiv$)(lgn{)}^{lgn}$

$($xv$)(\frac{3}{2}{)}^n$

$($xvi$)lg{n}^5$

$($xvii$)n{2}^n$