I'm not sure about this response, but am 80% confident this is true.

If you split the region of integration for f(x) into n rectangles, starting from 0 and ending at 1, the area is . Now i is somewhere inbetween the beginning of the rectangle and the end, but as n goes to infinity any method of finding i will work.

So let Then I believe this leads to the conclusion that your limit is the same as: