You can certainly avoid all the tedium of the epsilon-delta arguments while mastering calculus. In fact there were over a hundred years between the invention of calculus and the formalization of the epsilon-delta definition! Calculus was not invented in the same way it is taught, and the early masters of calculus were much more intuitive than rigorous.
For references, calculus was first published in 1684:
You could perhaps try to learn calculus the way it was taught three hundred years ago, but I'm pretty sure it would be more difficult, not easier, than the standard presentation. In the early years calculus was considered black magic only the smartest few could master, probably not unlike the way, say, String Theory is seen today.
For references, calculus was first published in 1684:
http://en.wikipedia.org/wiki/History_of_calculus
But the epsilon-delta definition wasn't formalized until 1817:
http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_...