any calculi without limits?

lolgarithms

do you know of any nonstandard calculi that does not have to use limits or the standard part function? I don't really care how weird it is, as long as you can define the derivative and the integral.

Civilized

Smooth infinitesimal analysis might be what your looking for, a nonstandard model in the logicians sense of true infinitesimals along with derivatives and integrals, but without limits and without any standard part functions ala robinson. The price to pay is that the law of the excluded middle does not hold in SIA, and in general it is not equivalent to standard analysis.