Why are derivatives defined before formal discussion of limits?
In a fully formal development, a full treatment of limits would of course come first. We prefer, though, to first introduce limits (and other concepts, for that matter) by showing them in action where the need for limits arises naturallyin finding derivatives. So we first introduce limits informally, illustrating their meaning and uses in context. We study limits in their own right after a section or two of using limits in context. We think that students benefit from having some sandbox time with limits before treating them formally.
