Using Limits to Determine Asymptotic Relationship
Knowing $3^4=81$ and $4^3=64$, we have $f(n)=81^n$ and $g(n)=64^n$. Using this we obtain:
$$\lim\limits_{n \to \infty}\frac{f(n)}{g(n)}=\lim\limits_{n \to \infty}\left(\frac{81}{64}\right)^n=+\infty$$
so you have your &quot;Note 2&quot;. By definition last limit mean $\forall C&gt;0$ we have  $\exists N \in \mathbb{N}$, such that $\forall n&gt;N$ holds $\frac{f(n)}{g(n)} \geqslant C$ so we have $f\in \Omega(g)$.
For further let's assume, that we have positive functions and $g$ have not $0$s subsequence.
Following question. Your &quot;note 1&quot; means that $\lim\limits_{n \to \infty}\frac{f(n)}{g(n)}$ is not $0$ and is not $\infty$, but equals to some number between them.
And last. Suppose $\lim\limits_{n \to \infty}\frac{f(n)}{g(n)}=A&gt;0$. This means, that $A-\varepsilon&lt;\frac{f(n)}{g(n)} &lt; A+\varepsilon$, for appropriate $n$, and so we can conclude $f \in \Theta(g)$.