PART $2-$ Comparing the orders of growth between two algorithms $(4$pts$)$

Say we have two different algorithms with respective runtimes of f$($n$)$ and g$($n$).$ Given the following cases, prove whether or not f$($n$)$ is O$($g$($n$))$ in each case.

You need to show your work with the crucial steps, e$.$g$.$ finding the necessarily constant c in the definition of Big$-$O$.$ You may utilize the pull$-$up$/$pull$-$down or the l'Hopital's trick discussed in class whenever needed.

You will be penalized for up to $1$ pt in total if all your work for this entire PART exceeds a whole A$4/$US letter page in $12$px font size $($assuming you start from the top of the page$).$

P$.$S$.$ sqrt$($n$)$ means the square$-$root of n$,$ aka n$^((1)/(2)).$

Case

f$($n$)$

g$($n$)$

A

log$($n$1000)$

log$($n$0\mathrm{.}0001)$

B

sqrt$($n$)$