  In math.QA/0309252, the author proved a number of multivariate elliptic
hypergeometric integrals. The purpose of the present note is to explore more
carefully the various limiting cases (hyperbolic, trigonometric, rational, and
classical) that exist. In particular, we show (using some new estimates of
generalized gamma functions) that the hyperbolic integrals (previously treated
as purely formal limits) are indeed limiting cases. We also obtain a number of
new trigonometric (q-hypergeometric) integral identities as limits from the
elliptic level.
