Put functions in ascending order by their growth rate

Question

I need to put the given functions in an ascending order by their growth rate. So that every function is $O(other- function)$ or to show they are equal in growth rate.

The functions are as follows:

$f_1(n) = n^2$

$f_2(n) = 4^\sqrt{n}$

$f_3(n) = \frac{n^3}{\log_{2}n}$

$f_4(n) = 100n^2$

$f_5(n) = n\log_{2}^2n$

As I understand, I need to find the limits of ratios of these functions which are equal to 0. However, I am not sure which functions I should be comparing and I don't really know how to find these limits.

Any help would be very appreciated!

Thank you in advance!

Answer 1

Hint:

Using that $1<\log n<\sqrt n<n$, where $<$ is understood as "grows slower than", we have

$$n<n\log_2^2n<n^2<\frac{n^3}{\log_2n}<n^3$$