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.

impossibleto obtain in real analysis. You could take the complex-analysis proof and rewrite everything in terms of real numbers (real and imaginary parts) without ever mentioning complex numbers. Of course those proofs would be ugly, hard to find and hard to understand, but they would not be impossible. Similar comments apply to nonstandard analysis. $\endgroup$ – Andreas Blass Apr 5 '15 at 19:41