Generically speaking, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. However, it is likely safe to say that of the most common meaning for the term "Non-Standard Analysis" is a rigorous formulation of Analysis using infinitesimals, as an alternative to the analysis we all know in the -δ formulation.Suppose a freshman is questioned on the derivative dy dx for the cubic function y = x 3 . Then after the difference-quotient is found,he often just says 'substituting by ∆x = 0', dy dx = 3x 2 . And the instructors often have trouble explaining the difference between '∆x set to 0' and 'lim ∆x→0 '.