What is the limit as x approaches infinity of lnx?

First of all it is important to say that $\infty$, without any sign in front of, would be interpreted as both, and it is a mistake!
The argument of a logarithmic function has to be positive, so the domain of the function $y = \ln x$ is $\left(0 , + \infty\right)$.
${\lim}_{x \rightarrow + \infty} \ln x = + \infty$, as shown by the graphic.