Logarithm Tends to Negative Infinity
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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:
\(\ds \ln x\) | \(=\) | \(\ds \int_1^x \dfrac 1 t \ \mathrm dt\) |
The result follows from Integral of Reciprocal is Divergent.
$\blacksquare$