Can the homographic function:
$$f(x)=\frac{1+\frac{x}{a}}{1-\frac{x}{1-a}}$$
where a ∈ (0,1), be approximated by an exponential function for the interval x ∈ [0,1-a] (where the function f(x) behaves as an increasing and convex function)? Graphically, the exponential function $exp(\frac{1}{a(1-a)}x)$ seems to approximate the function f(x) better than the Taylor approximation (whatever degree is chosen) for that interval, but can one formally find an exponential function that is a good approximation to the function f(x)? I know that an exponential function can be approximated to a rational function using the method proposed by Padé, but my problem is actually the opposite.