I think the key missing insight for me was that a continuous process and a discrete process can both point to the same result.
A pixellated word on a screen ("cat") conveys the same meaning as a perfectly smooth vector of the same word. The math idea, to me, is "can a discrete description/process" point at the same result that we get from a continuous one?
The idea of limits, is essentially figuring out when a discrete epsilon can still lead us to the true value of f(x) (if this trick works, we call it a continuous function, and can use calculus with it.)
Unfortunately, I doubt that these implications will be discussed in your average Mathematics classroom. Actually, my experiences with mathematics in primary, secondary and a lot of tertiary education, both as a student, teacher and observator, leads me to believe that conceptual discussion is rare, whilst procedural discussions (how to perform some calculation) are plenty and abound.