I teach at a community college, and I love teaching calculus. I hope my course is at least as good as university courses. I think it is a big benefit to introduce them to the deep logic of the epsilon-delta definition of limits, but I do not prove anything with them, and don't think that's very valuable for beginners.
I think any attention to limits at the beginning of the course is not wise. Limits are strange new creatures for the students, and at the beginning of the course they have no idea why this weirdness is necessary. For most students it will be alienating to spend time there before they've learned about derivatives.
So I start with a full unit on understanding derivatives as rate of change, slope, velocity, ... (Anton doesn't have nearly enough of what I want, so I use lots of material from Boelkins' free text for this part.) We use the regular limit notation, but I just say that it means infinitely close. (And I say that that's not really enough. But it's about what Newton did, and mathematicians took 150 years to do better, so it's good enough for a start.)
Then, after the units on rules of derivatives (I call them properties) and applications, once the students see how valuable derivatives are, I introduce limits more formally. I do not ask any test questions regarding limits, but I do introduce the epsilon-delta definition.
I call it the sentence that took 150 years to write, and I write it almost like a poem, one short phrase on each line:
lim (x->c) f(x) = L
for any ε > 0
there is a δ > 0
if 0 < abs(x-c) < δ
then abs(f(x)-L) < ε.
I draw diagrams with colorful bands, much like what you see in any good textbook, and explain with lots of pointing.
Then I give a colorful example to help the students understand the machinery. (I got it from a colleague many years ago. Thanks, Diane.) It uses a cookie crispness index on the y-axis, and the time your cookies are in the oven on the x-axis. Before I get into it, I ask the students what their favorite cookies are and how crisp they like them on a scale from 0 to 10, where 0 is doughy, 3 is still very soft and chewy, 6 is getting more crispy than chewy, and 9 is starting to get burned. Then I use one student's preference for my example.
I draw a line for the function that connects cooking time to crispness, and from that I estimate the time I need to bake. Then I play with them about how perfect the cookie will be (ε). Now I get to draw my bands. If I want the cookie that close to perfect, how close to perfect must my timing be, when I pull that tray out of the oven?
No testing, no proofs, just a good example to chew on. ;^)