sure. and despite all the trouble people have, their definition is quite simple. the crux of the problem is understanding continuity. if you truly understand the continuity of the reals (which is the "real" invention), limits can be derived almost effortlessly.
in most calculus teaching, continuity is quickly stated and then there's a bunch of rules and these seemingly contradictory discussions involving delta and epsilon. they're not contradictory if continuity is understood and internalized.
Do you prefer the Dedekind cut construction of the real numbers over the identification of Cauchy sequences of rational numbers with common limit? In ZFC, they coincide, but in constructive mathematics, one can build a model wherein they differ (see https://mathoverflow.net/questions/128569/a-model-where-dede... ).
