Is is true that $\lim(\ln x) = \ln(\lim x)$? As the title says, I want to ask everyone if $\lim(\ln x) = \ln(\lim x)$ when x approach to infinite with any function.
The function $\ln x:\mathbb{R^{+}}\to\mathbb{R}$ is a continuous function. So $\forall x_0 \in \mathbb{R^{+}} \ \ \lim_{x\to x_0 }\ln x =\ln x_0=\ln\lim_{x\to x_0}x$.
$$\lim_{x\to\infty} \ln(x)=\infty$$
$$\ln\left(\lim_{x\to\infty}x\right)=\infty$$
If I understood what you were saying properly, it was a bit unclear..