Why is the definition of "limit" difficult to understand at first? Tomorrow I teach my students about limits of sequences.  I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier.  But first I need to know, why would the definition be difficult to understand?  Too many logical quantifiers?  The definition of "sequence" itself is vague?  The usual examples are not enough to develop intuition?
I hope this question is not too vague.

This answer is a mere recall from my first exposure to calculus from a horrible, condescending instructor whom made calculus the most difficult subject for me. I am certain now though, that I am better than her. 
That aside, the reason limits were so difficult are the following
(i) Calculating limits requires many basic foundations which most students lack, these includes absolute values, distributive properties of polynomials, knowing certain properties of functions (trig, exponential, rational functions, etc...). Most exercises involve combining everything they learned in pre-calculus which most lack 
(ii). Other reason is because calculus is the first time they will have to look at math differently. I mean I wasn't very comfortable dealing with the infinite, it was fine and dandy counting finiteness, but infinity! It's also difficult to accept infinity as a mere concept rather than a thing, I blame this on the rubbish they broadcast on the science channel to fool the uneducated. 
