  We study the asymptotic behaviour of Bessel functions associated of root
systems of type $A_{n-1}$ and type $B_n$ with positive multiplicities as the
rank $n$ tends to infinity. In both cases, we characterize the possible limit
functions and the Vershik-Kerov type sequences of spectral parameters for which
such limits exist. In the type $A$ case, this gives a new and very natural
approach to recent results by Assiotis and Najnudel in the context of
$\beta$-ensembles in random matrix theory. These results generalize known facts
about the approximation of the (positive-definite) Olshanski spherical
functions of the space of infinite-dimensional Hermitian matrices over $\mathbb
F = \mathbb R, \mathbb C, \mathbb H$ (with the action of the associated
infinite unitary group) by spherical functions of finite-dimensional spaces of
Hermitian matrices. In the type B case, our results include asymptotic results
for the spherical functions associated with the Cartan motion groups of
non-compact Grassmannians as the rank goes to infinity, and a classification of
the Olshanski spherical functions of the associated inductive limits.
