Little O notation relationship Given the functions $()=^{n}$ and $()=10^{10n}$, I am trying to establish the following relationship: $()\notin o(())$.

I know to show for the opposite, $()\in o(())$,  I would need to choos $c$ and $n_{0}$ such that $\exists$ $c$, $\exists$ $n_{0}$, $\forall n\geq{n_{0}}$ then $f(n)\leq c. g(n)$, but how should I choose $c$ and $n_{0}$. Note that I am beginner in studying CS and any help would be greatly appreciated.
One possibility is to merely apply the definition. That is, we see that if $\lim_{n \to \infty} f(n) / g(n) = 0$, then $f(n) = o(g(n))$. Computing this, we have that $$\lim_{n \to \infty} f(n) / g(n) = \lim_{n \to \infty} n^n/10^{10n} = \infty \neq 0.$$ We conclude that $f(n) = o(g(n))$ does not hold.