Is $ \lim_{\epsilon \to 0^+} \int_0^\epsilon \frac{1}{x} dx $ well-defined?
I've read about Cauchy convergence, and I can show that the integral is not Cauchy convergent, but I'm not sure if that is applicable to this problem.
This is not needed to answer this question, but if the readers are interested in the background to my question: An integral of this form is arising when when I use Galerkin's method to solve an integral equation involving an integrand that goes as $\frac{1}{\text{radius}}$ in $2$-dimensions. If I assume that the above integral is just $0$, then I get answers that are physically correct for instances where I can compare against a known solution. But I'd really like to understand if there is a more mathematically rigorous way of handling this limit.
EDIT: Thanks everyone for their suggestions so far. If the limit is not a real number, I may need to back up and question my assumptions. I arrived at this this limit of an integral by doing an integration by parts in 2-dimensions over an area $A$:
$$ \iint_A f(x,y) \nabla \cdot \vec g(x,y) dA \,=\, \oint f(x,y) \vec g(x,y) \cdot \hat n dl \,-\, \iint_A \nabla f(x,y) \cdot \vec g(x,y) dA $$
where $\hat n$ is the unit outward normal of the area, and $dl$ is an infinitesimal step around the area's contour.
In some cases in my geometry, the resulting contour encloses no area, so I assumed that I should take the limit of this integral as the contour radius goes to 0. (These contour radii of zero occur because I am solving the integral equation on the surfaces of bodies of revolution, and some of my basis functions terminate at the axis of revolution.) But this does not give a meaningful limit. Maybe simply saying that the contour integral must be zero around an area of zero is the best that I can do, but that seems hand-wavy to me. I've only had up through undergrad analysis, so I'm not sure if this "define it to be 0" is mathematically justifiable, but I get the correct physical answer. If anyone can offer anything else, I will read it with interest. Thanks!