How can I find which is the most appropriate symbol (O, o, $\Omega$, $\theta$, $\omega$) to describe $f(n)\in$__ $g(n)$?

Question

when the given functions are $f(n) = \log^3(n^{10})$ and $g(n) = n^{1/4}$

I usually use limits but in this case the limit doesn't exist.. How do I go about figuring out which notation to use? :(

Answer 1

we can write $f(n) = 10^3\log^3(n) = c_1 \log^3(n)$ and $g(n) = n^{c_2}$. as growth of $\log(n)$ is less than $n$, we can found that $f(n) = o(g(n))$ (little $oh$). In the other words, as we can write $g(n) = (e^{0.1\times f(n)^{1/3}})^{1/4}$, growth of $g(n)$ is strictly greather than $f(n)$. Hence, $f(n) = o(g(n))$