I have a textbook question where I have to rank 5 functions by increasing order of growth. I won't post the functions that I need to solve as I am not looking for the answer but a different method.
I know how to compare two functions $f$ and $g$ and determine if one grows faster, smaller or if they grow at a rate where one is faster by the multiple of some constant (O, o, $\Omega$, $\omega$ and $\Theta$) by using limits like so:
$$\lim_{n->\infty} \frac{f(n)}{g(n)}$$
where whether the value of limit is $0$, some non-zero constant or $\infty$ determines if and how $f$ is bounded by $g$.
This would involve 10 comparisons to compare each function (I'm pretty sure). I know this is a funny question since this is for an algorithms course, but is there a more efficient way to do this so that it will not take up 3 pages in my work book?
Growth rates are transitive, in the sense that if $f=o(g)$ and $g=o(h)$, then $f=o(h)$. So once you work out which order the functions are sorted in, you can just compare each function to the next-largest function.