1. The problem is to test the continuity of the function $f(x) = \ln(x)$ at $x=0$. 2. Recall that the natural logarithm function $\ln(x)$ is defined only for $x > 0$. 3. To test continuity at $x=0$, we need to check if $\lim_{x \to 0} f(x) = f(0)$. 4. However, $f(0)$ is not defined because $\ln(0)$ is undefined. 5. Next, consider the limit from the right side: $$\lim_{x \to 0^+} \ln(x) = -\infty.$$ This means the function tends to negative infinity as $x$ approaches 0 from the right. 6. Since $f(0)$ is undefined and the limit does not approach a finite value, $f(x) = \ln(x)$ is not continuous at $x=0$. Final answer: $f(x) = \ln(x)$ is not continuous at $x=0$ because it is not defined there and the limit does not exist finitely.