Getting a specific digit from a ratio expansion in any base (nth digit of x/y)

Is there an algorithm that can calculate the digits of a repeating-decimal ratio without starting at the beginning?

I'm looking for a solution that doesn't use arbitrarily sized integers, since this should work for cases where the decimal expansion may be arbitrarily long.

For example, 33/59 expands to a repeating decimal with 58 digits. If I wanted to verify that, how could I calculate the digits starting at the 58th place?

Edited - with the ratio 2124679 / 2147483647, how to get the hundred digits in the 2147484600th through 2147484700th places.