I am having a problem in understanding the $\epsilon$ - $\delta$ definition of limit of a function. Actually, I understand what the definition says perfectly, that, no matter how close we get to $L$ if we can find an interval around $c$$(x\rightarrow c)$ for which every $f(x)$ is in that interval around $L$, then $L$ is called the limit as $x$$\rightarrow$$c$.
The problem is, I'd say, I don't have any solid feel or comprehensive understanding for what the definition is really trying to say. I avoided saying "intuitive understanding" because then people comment like "Everyone's intuition is different", "Forget intuition, just consider the definition", etc. I'm just not having the total clarity that one has after studying any mathematical concept.
I haven't studied anything about sequences or the limit of a sequence concept, which appears before the limit of a function definition in many famous real analysis textbooks. I wonder if this is the reason behind the fuzziness.
So, it would help if you clarify. If you've two cents to directly attack the question of better understanding the $\epsilon$ - $\delta$ definition, please go ahead.