Btw one thing worth noting if you really do want to work with surreals is that it may be more productive to think in terms of { stuff | stuff } rather than limits. (Similar to my "break ties by least birthday" suggestion.) Sequences don't have limits in the surreals, but there is nonetheless a theory of surreal exponentiation based on {stuff | stuff}. Integration... well, it's less obvious to me that integration based on limits should fail, but if it does, you could try to do it based on {stuff | stuff}. (The existing failed attempt at a theory of surreal integration takes that approach, though as I said above, that's not really the same thing, as that's for functions with the surreals as the domain.)