Modelling Bounded Utility Functions
There are many such utility functions. Most commonly: 

\begin{equation}
u(x)=L-\mathrm e^{-ax},\tag{1}
\end{equation}
where typically $a&gt;0$. Or 
\begin{equation}
u(x)=L-(x-a)^2.\tag{2}
\end{equation}

Both of these are uniformly bounded below $L\in\mathbb R$ and both tend to $L$ as the functions approach their respective maximum.