Keisler has a introductory text book (available free here: https://www.math.wisc.edu/~keisler/calc.html https://www.math.wisc.edu/~keisler/calc.html) that uses

Keisler has a introductory text book (available free here: https://www.math.wisc.edu/~keisler/calc.html https://www.math.wisc.edu/~keisler/calc.html) that uses non-standard analysis, which he claims is easier for beginners to understand than the notorious delta-epsilon proofs. I can't say how effective it is at that goal, but it is all mathematically legitimate, and has a nice intuitive appeal to it.