  The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi
functions in one variable and include the spherical functions of non-compact
Grassmann manifolds over the real, complex or quaternionic numbers. There are
various limit transitions known for such hypergeometric functions. In the
present paper, we use an explicit form of the Harish-Chandra integral
representation as well as an interpolated variant, in order to obtain limit
results for three continuous classes of hypergeometric functions of type BC
which are distinguished by explicit, sharp and uniform error bounds. The first
limit realizes the approximation of the spherical functions of infinite
dimensional Grassmannians of fixed rank; here hypergeometric functions of type
A appear as limits. The second limit is a contraction limit towards Bessel
functions of Dunkl type.
