I'm trying to figure out how to prove the following but to no avail.
Given the following functions :
$f(n) = n^3 -4n$
$g(n) = 5n^2 + 3n$
I have to show that $g(n) = o(f(n))$ by definition, that is not using limits or O definitions.
Notice that we say that $f(n)$ is $o(g(n))$ if for any real constant $\epsilon > 0$, there exists an integer constant $n_0 >= 1$ such that $f(n) < \epsilon g(n)$ for every integer $n>=n_0$.
I was trying to use some calculus techniques to solve it.
Here is what I have so far:
I'm looking for $n_0 >= 1$ such that $\epsilon_0 >= (5n^2+3n)/(n^3-4n)$ for every $n>= n_0$
Can someone explain me how can I continue from here? If I solve this inequality I believe It will give me the required $n_0$ for the $\epsilon$.