There are vast sections of mathematics which cannot be understood without first understanding limits. There are very few areas of mathematics which require understanding infinitesimals.
Sure. I just think the infinitesimal is an interesting approach.
Fancy that! I'm at the UW right now. I have been considering a few different routes. Philosophy, Classics, Music, Computer Science, or Math. I'd really like to go for a Math undergrad with a minor in one of the other subjects, but I'm going to need a supplement during the summer to catch up with some of the other math guys at the UW. Lots of competition.
Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.