For some reason, students I teach always love epsilon-delta (not that they write good epsilon-delta proofs per se), and the more "wrong" I teach it, the more they enjoy it. The "wrong" thing that I like to do is to define real numbers via Cauchy sequences right at the beginning, at least in a hand-wavy way.

Calling a real number "real" is Orwellian, really- none of you have ever seen a real number. You might estimate pi to 100 or 1000 or a million places after the decimal point, but you can never write it all down- you never know precisely what it is. Even numbers like 0 and 1 are unknown as real numbers. You can write "0.00000..." until you're blue in the face, but you'll never know if the number you've written was actually zero or not, because there might be a sneaky "...000001..." coming up just around the corner to bite you.

So real numbers come with "fuzz". They're inherently "fuzz". There's no way around it. You can pretend they're points on a line, and crash face-first into Zeno's paradoxes, or you can accept their fuzziness and work with it; and I argue that calculus is none other than that "second approach".

So my "epsilon" is the fuzz. It's all those digits of "pi" (or of "zero") you never wrote down. It's the tree falling when nobody is there to hear it. It's the gap between human knowledge and universal truth. As such, it's part of what a real number is- a real number (as observed by mortal beings) comes with fuzz. A continuous function from the reals to the reals, then, has to "respect the fuzz".

Anyway, counter-intuitive though it is (perhaps), this motivation has worked wonders for me in practice. And so I keep using it, and keep getting excited about it, and it's the best part of the course, year after year.

**Added**: Interpreting the verb "to motivate" in another way, I always discuss the history of the ideas in some depth (I learnt it myself from wikipedia and books on the history of mathematics), and just how much people struggled to find the "right" definition, with no success, until Bernard Bolzano (primarily a Catholic priest!) finally hit upon an idea that worked in 1810. What idea were they trying to capture? Why was it so hard? How come it took 2500 years (Zeno of Elea to Bolzano) to find the right idea?

I'll also discuss the definition having been reworked and distilled by many many people- first its inventors, then mathematicians, then textbook writers, becoming more and more refined and smaller and smaller until that which is left looks to one who sees it for the first time like a small cold hard stone. It's only once you polish it (working it over in your mind, and solve problems) and shine it under a bright light (make sense for yourself of all those nested quantifiers) that you can finally see it for what it is- a diamond.