Integral of Reciprocal is Divergent

Theorem

Unbounded Above

$\displaystyle \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$

To Zero

$\displaystyle \int_\gamma^1 \frac {\d x} x \to -\infty$ as $\gamma \to 0^+$

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

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