A limit exists iff and only the left limit and the right limit exist and are equal to each other

Actually, what you wrote is not well known, and is in fact not even true.

What is well known is the fact that

Let $f$ be a function $\color{red}{\text{defined on some open neighborhood of $a$}}$. Then, $$\lim_{x \to a} f(x) = L \iff \lim_{x \to a+}f(x) = L = \lim_{x \to a-}f(x)$$

Always read the fine print.