Without the reals you can't assume that all limits exist.  The reals are the only totally ordered complete field.  As such, whether they're "practical" or "useful" is irrelevant.  They are a mathematical object worthy of study in their own right.<p>Calculus is easier with the reals than without.  The computing you do is an approximation to the reals - trying to do numerical analysis without the reals is horrendous.  I've seen so-called "constructivists" attempting it, and I'd rather work with the reals any day.