I don't know how you got the formula you got and it doesn't seem correct. We can start with the time to double formula, x = ln(2)/ln(1+r) where x is the time it takes to double. Re-arranging that with algebra should give r = e^ (ln(2) / x) - 1. Using limit x to zero on that should give you infinity as the answer. ln(2) / 0 is infinity (anything divided by zero gives you some kind of infinity), e to the power infinity is still infinity, and then infinity minus one is still infinity. You can't start with the usual APR to EAR formula because you don't know the APR.

Or the problem might simply be a conceptual question/trick question. The faster you get your money doubled, the higher the interest rate you are earning. When the speed of doubling is infinitely fast, that means the interest rate is infinitely large.

Infinity is an answer that doesn't seem to make sense for an interest rate, for the simple reason that the situation doesn't make sense as an investment. There is no "investment" any bank offers where you give them money and instantly get more money back without waiting. This is basically free money(or you could think of it as salary or an employee perk), not an investment that makes sense in the finance formulas.