expressing repeating fractions as integral of 1/x

EDIT: Nevermind about the "fraction" part. What I mean is a number whose last digit repeats infinitely.

The function $\displaystyle f(x)=\frac{1}{x}$ is not defined at x=0.

What is the proper way to express numbers with an infinitely repeating last digit as the area under this function?

Taking a shot at it, if I take as an example the number $\displaystyle 8\overline{33333}$ I would say...

$\displaystyle \lim_{b \to 0 }\; \int_{b}^{1}\frac{1}{x}\; dx=8\overline{33333}$

Is this correct? Is there a less ugly way to do it?

Thanks