I asked a similar question yesterday, but I wasn't very clear and/or I don't understand how the answer given met my conditions.
I want a function that takes elements of $\mathbb R$ to other elements of $\mathbb R$ such that:
(i) For every $x \in \mathbb R, f(x) < x$, and $f(x) \in \mathbb R$
(ii) $f(x)$ asymptotically approaches some chosen positive limit $L$ as $x \to \infty$
One answer given was that I could take the function $f(x) = ex^{-x}$ and then use $g(x) = f(x - L) + L$ as my function. However, I don't see how, for instance, if my limit is 100 and x = 10, it satisfies those conditions.