  Let $F_{BC}(\lambda,k;t)$ be the Heckman-Opdam hypergeometric function of
type BC with multiplicities $k=(k_1,k_2,k_3)$ and weighted half sum $\rho(k)$
of positive roots. We prove that $F_{BC}(\lambda+\rho(k),k;t)$ converges for
$k_1+k_2\to\infty$ and $k_1/k_2\to \infty$ to a function of type A for $t\in\b
R^n$ and $\lambda\in\b C^n$. This limit is obtained from a corresponding result
for Jacobi polynomials of type BC, which is proven for a slightly more general
limit behavior of the multiplicities, using an explicit representation of
Jacobi polynomials in terms of Jack polynomials. Our limits include limit
transitions for the spherical functions of non-compact Grassmann manifolds over
one of the fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H$ when the rank is
fixed and the dimension tends to infinity. The limit functions turn out to be
exactly the spherical functions of the corresponding infinite dimensional
Grassmann manifold in the sense of Olshanski.
