Integral of Reciprocal is Divergent/To Zero

Theorem

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

Thus the improper integral $\displaystyle \int_{\to 0^+}^1 \frac {\d x} x$ does not exist.

Proof
Put $x = \dfrac 1 z$.

Then:

But as $\gamma \to 0+$, we have that $\dfrac 1 \gamma \to +\infty$.

Hence, from Integral of Reciprocal Unbounded Above is Divergent:
 * $\displaystyle \int_1^{1 / \gamma} \frac {\d z} z \to +\infty$

as $\dfrac 1 \gamma \to +\infty$.

The result follows.