Asymptotic Relationship from Limit
The theorems you quote are unfortunately wrong. As an example, $2 + \sin n = \Theta(1)$ although the limit $\lim_{n\to\infty} \frac{2+\sin n}{1}$ doesn't exist.

Here are some theorems which do hold.

Theorem 1. Let $f,g$ be two functions such that $f(n),g(n)$ are eventually positive. If the limit $c := \lim_{n\to\infty} f(n)/g(n)$ exists then


$f(n) = O(g(n))$ iff $0 \leq c &lt; \infty$.
$f(n) = \Omega(g(n))$ iff $0 &lt; c \leq \infty$.
$f(n) = \Theta(g(n))$ iff $0 &lt; c &lt; \infty$.
$f(n) = o(g(n))$ iff $c = 0$.
$f(n) = \omega(g(n))$ iff $c = \infty$.


Theorem 2. If $f,g$ are two polynomials which are eventually positive, then the limit $\lim_{n\to\infty} f(n)/g(n)$ exists.

Theorem 3. Let $f,g$ be two functions such that $f(n),g(n)$ are eventually positive. Then


$f(n) = O(g(n))$ iff $\lim\sup_{n\to\infty} \frac{f(n)}{g(n)} &lt; \infty$.
$f(n) = \Omega(g(n))$ iff $\lim\inf_{n\to\infty} \frac{f(n)}{g(n)} &gt; 0$.
$f(n) = o(g(n))$ iff $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 0$.
$f(n) = \omega(g(n))$ iff $\lim_{n\to\infty} \frac{f(n)}{g(n)} = \infty$.