I know, I know, this topic has already been beat to death, but I'm still having a hard time understanding it despite having already read several forum threads and educational articles. Intuitively, the definition is stating that no matter how narrow we choose to make the "epsilon band" surrounding f(x) = L, there will always be a "delta band" surrounding x = a that contains the x-values that correspond to all the f(x) values contained within the epsilon band. What this is effectively saying is that the function curve is defined no matter how close it gets to x = a. Am I at least partly correct? Then, when we get down to the nitty gritty rigorous definition and define delta in terms of epsilon, is that the equivalent of proving that the limit exists? Is the fact that there exists a relationship between delta and epsilon that satisfies the definition proof enough that the limit exists? I'm searching for a bridge that connects the intuitive definition with the rigorous definition but having I'm having a difficult time finding one. Basically, what I'm asking is: How exactly does the epsilon-delta ordeal prove that the limit exists?