Integral of Reciprocal is Divergent

To Zero
Thus the improper integrals $\ds \int_1^{\to +\infty} \frac {\d x} x$ and $\ds \int_{\to 0^+}^1 \frac {\d x} x$ do not exist.

In particular: $\ds \int_{\to 0^+}^{\to +\infty} \frac {\d x} x$ certainly does not exist

Also see

 * Logarithm Tends to Infinity