Finding an analytic form of a function that satisfies asymptotic conditions

Question

I have a family of functions that I obtain numerically. They depend on $x$ and parametrically also upon a certain parameter $L$. I would like to find an analytical form for this family of functions so that I can fit certain parameters to the numerical data and obtain the (best) analytical form of $f(x;L)$. These functions are only defined within the interval $x = 0$ to $x = \frac{L}{2}$. I know certain exact properties of my functions:

1. As $x \rightarrow 0$, $f(x;L) \rightarrow \frac{A(L)}{x}$.
2. As $x \rightarrow \frac{L}{2}$, $\partial_x f(x;L) \rightarrow 0$.
3. In between $x = 0$ and $x = \frac{L}{2}$ the functions should decay exponentially, as $B(L) \, e^{-\frac{x}{C(L)}} + D(L)$.
4. $f(x;L)$ and its derivative $\partial_x f(x;L)$ must be continuous within the interval $x \in (0, \frac{L}{2})$.

I have tried with different combinations of exponentials and so on, but I'm stuck finding a suitable form. The numerical data look like this:

I am looking for help in finding the explicit dependence of $f$ upon $x$ after which I can use the data to find numerical (possibly polynomial) fits for the dependence of $A, B, C$ and $D$ upon $L$.

I apologize in advance for bad terminology etc., it's been a while since my last formal lessons in mathematics!