I wonder whether non-standard analysis, non-archimedean extensions of reals such as surreal or hyperreal numbers can help us in obtaining standard-analytic results which are not possible to obtain by means of standard analysis?
Finding limits in closed form which are impossible to find in standard analysis
Generalizing functions to the values where standard analysis does not provide any satisfactory method
Eliminating undetermined forms where it is impossible by standard-analysis means
Finding identities and relations between standard numbers and functions that are inpossible by standard means
Or non-standard analysis is just as powerful as epsilon-delta method in regard of standard numbers?
I am asking this because evidently complex numbers gave a lot of new powerful methods and results - from solving algebraic equations towards Fourier transforms, integral forms for finite differences, number theory etc.