Nonexistence of a limit What does it mean for limit not to exist?

By negating the definition of limit we have:
$$\exists \epsilon &gt; 0, \quad \forall \delta &gt;0, \exists x \in I \setminus\{c\}$$ such that $0 &lt; |x - c| &lt; \delta$ but $|f(x) - L| \geq \epsilon$.

I understand what this means. But what does $L$ mean here? $L$ does not even exist and in fact we are using it as a part of the definition.  Can someone clarify?
The definition of limit should start with $\exists L$. Consequently the negation should start with $\forall L$
$L$ is your candidate for the limit of your function. This does not mean that your function $f$ does not converge, but it does mean that $L$ is not the limit it's approaching (whether $f$ does converge or not is irrelevant). But if you take into consideration EVERY real number $L$ and then negate the definition it means that $f$ diverges as there is no limit $L$.