Logarithm Tends to Negative Infinity

Theorem
Let $x \in \R$ be a real number such that $x > 0$.

Let $\ln x$ be the natural logarithm of $x$.

Then:
 * $\ln x \to -\infty$ as $x \to 0^+$

Proof
From the definition of natural logarithm (or from Equivalence of Definitions of Natural Logarithm):

The result follows from Integral of Reciprocal is Divergent.

Also see

 * Logarithm Tends to Infinity