  In this paper we derive non-classical Tauberian asymptotic at infinity for
the tail, the density and the derivatives thereof of a large class of
exponential functionals of subordinators. More precisely, we consider the case
when the L\'evy measure of the subordinator satisfies the well-known and mild
condition of positive increase. This is achieved via a convoluted application
of the saddle point method to the Mellin transform of these exponential
functionals which is given in terms of Bernstein-gamma functions. To apply the
saddle point method we improved the Stirling type of asymptotic for
Bernstein-gamma functions and the latter is of interest beyond this paper as
the Bernstein-gamma functions are applicable in different settings especially
through their asymptotic behaviour in the complex plane. As an application we
have derived the asymptotic of the density and its derivatives for all
exponential functionals of non-decreasing, potentially compound Poisson
processes which turns out to be precisely as that of an exponentially
distributed random variable. We show further that a large class of densities
are even analytic in a cone of the complex plane.
