Answer
i) Increasing in $(0,\infty)$
ii) Never decreasing in the domain of f
iii) The tangent is never horizontal.
Work Step by Step
$f(x)=\ln{x}$
Let $f:I\to R$ be differentiable on the interval $I$, then $f(x)$ is increasing if and only if $f'(x)\ge0 \hspace{0.2cm}\forall x\in I$ .
$f'(x)=\begin{cases} \frac{1}{x}>0;\hspace{0.2cm} x>0\\\frac{1}{x}<0;\hspace{0.2cm}x<0\end{cases}$
This implies that $f(x)$ is increasing on $(0,\infty)$ and decreasing on $(-\infty,0)$. But the logarithmic function is only defined on the domain $(0,\infty)$, i.e. $I=(0,\infty)$. Therefore, we can conclude that $\ln{x}$ is an increasing function throughout its domain.
$f'(x)$ is not defined at $x=0$ and $f'(x)\neq 0 $ for any $x$. This means that the tangent is never horizontal. The tangent is horizontal at the critical points.
