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.
As far as I see it, there are two massive obstacles that a classroom must overcome when trying to come to grips with formal limits.

A big problem with understanding the limit definition, as you have pointed out, is the problem of quantifiers. There are so many, and the order in which they are employed is so important, that it can be difficult to keep track of. (And just when they think they understand, now they have to deal with uniform continuity, which to most will look exactly the same as continuity!)

To make matters worse, it's hard to describe a limit in a language that is precise enough to be useful to solve problems, but gentle enough to be swallowed easily. This is hardly surprising; literally thousands of years of mathematics was done before anyone really found the need to understand them completely.

A big problem with using the limit definition, is that it feels circular. This is often the first time students will have to consciously work ''forward'' to first convince themselves what the limit is before they can prove it.

For example, when you are solving $3x+4=5$ there is a mechanical list of procedures that one can go through to get to the answer. Even when doing something more technical, like verifying the trigonometric identity $\tan^2x + 1 = \sec^2x$, you can still start from what you're given and come up with the answer. Thoughtful, informed students will know this logic needs to be run in reverse to constitute a proof, but there is at least a foothold for one to grab onto; that original equation.

But when you are trying to find the limit of $x^2$ as $x\to 4$, if you don't have the intuition that the answer should be $16$, your work becomes very difficult. You can apply the limit definition to try to find a $\delta$, but the catch-22 is that you need to ``know'' what the limit is before you can even interpret which points are in the $\varepsilon$-neighborhood..
If you haven't done it already, please do look through Cauchy's rigorous definition of limits and continuity. If you are comfortable with the algebra of inequalities, the idea of limits becomes very much rigorous than intuitive. (Personally I found it a lot easier doing it the "Cauchy way")

See this. It discusses Cauchy's contribution to calculus and how he made calculus a rigorous mathematical field in a time where various (rather unknown) mathematicians logically (and beautifully) pointed various gaps in Newton's and Leibnitz's beautiful intuitive brain-child.