All of the calculus textbooks I've used (teaching at community colleges) start with the first chapter covering limits. (Perhaps after a review chapter.) I think this order is wrong.
Historically, Newton and Leibniz thought in terms of fluxions or infinitesimals. It took mathematicians 150 years to develop the logical (epsilon-delta) machinery of limits that puts calculus on a sound logical foundation. (And it turns out that this is not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way.)
The big ideas of calculus are derivatives for rate of change, and integration for areas and volumes. I think the course makes more sense for students if we start with "slopes of curvy lines" and velocity. So I'm bringing in outside material to help them explore the basic concept of the derivative from many perspectives. (Our textbook only has two sections that do this. Most only have one. I spend three weeks on it. Much of my material comes from Boelkins' Active Calculus.)
I give the limit definition of the derivative, but I say that for now we'll think of the limit as meaning that we get infinitely close. After we are done with simple derivatives, I go back to chapter one's limit exercises before doing trig derivatives, because we hit some unusual limits there (e.g. sinx/x).
I'd like to know why textbooks cover limits first. Is it just because the mathematicians writing them are stuck at a higher level, and want to first establish the validity of what will be done later? I don't think that's good pedagogy. I think calculus is beautiful and powerful, and that I can convey this better by waiting to introduce limits after students see why they need them.
Why cover limits first?