  We prove a limit relation for the Dunkl-Bessel function of type $B_N$ with
multiplicity parameters $k_1$ on the roots $\pm e_i$ and $k_2$ on $\pm e_i\pm
e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It
gives a good approximation in terms of the Dunkl-type Bessel function of type
$A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved
estimate is obtained from a corresponding limit relation for Bessel functions
on matrix cones.
