Modelling Bounded Utility Functions I'm trying to work out how to model a utility function that is bounded below some level. More precisely, given a specified limit $L$, I want to work out how to ensure that the utility of any outcome $o$ asymptotically approaches $L$ as the value of $o$ increases.

I'd really appreciate any help offered!
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.