How to carry out basis set fitting and extrapolation
If I well understand your question you have a set of experimental points $(E_1,X_1), (E_2,X_2), ... , (E_n,X_n)$. You want to fit to this set of points the function :
$$
E(X) = E_\infty + Ae^{-\alpha X}
$$
So, you want to approximately determine the parameters $E_\infty$ , $A$ and $\alpha$.
This requires a non-linear regression calculus. For example see https://mathworld.wolfram.com/NonlinearLeastSquaresFitting.html
This calculus is iterative and requires to set some &quot;guessed&quot; values of the parameters to start the iterative process.
Another method (much simpler but for approximative result only) is not iterative and doesn't require &quot;guessed&quot; values is explained in : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales
You could try it in folowing the numerical example below with $\quad y(x)=E(X)$ , $a=E_\infty$ , $b=A$ , $c=-\alpha$


Note : This simplified method is not reliable if the number of points is too low.