Infinite limit of ln(x) as x approaches 0 from the right

As $x$ approaches $0$ from the right, we plug in progressively smaller positive values such as $x = 0.1, 0.01, 0.001$, which give $\ln(0.1) \approx -2.303$, $\ln(0.01) \approx -4.605$, and $\ln(0.001) \approx -6.908$. These values are decreasing without bound, showing that $\ln(x)$ grows more and more negative as $x \to 0^+$.

This unbounded negative growth means the limit does not settle on a finite number, so we write $\displaystyle\lim_{x \to 0^+} \ln(x) = -\infty$. Since this one-sided limit is infinite and $f(x) = \ln(x)$ is undefined at $x = 0$, the vertical line $x = 0$ is a vertical asymptote of the graph.