  We investigate the Reshetikhin--Turaev invariants associated to SU(2) for the
3-manifolds M obtained by doing any rational surgery along the figure 8 knot.
In particular, we express these invariants in terms of certain complex double
contour integrals. These integral formulae allow us to propose a formula for
the leading asymptotics of the invariants in the limit of large quantum level.
We analyze this expression using the saddle point method. We construct a
certain surjection from the set of stationary points for the relevant phase
functions onto the space of conjugacy classes of nonabelian
SL(2,C)-representations of the fundamental group of M and prove that the values
of these phase functions at the relevant stationary points equal the classical
Chern--Simons invariants of the corresponding flat SU(2)-connections. Our
findings are in agreement with the asymptotic expansion conjecture. Moreover,
we calculate the leading asymptotics of the colored Jones polynomial of the
figure 8 knot following R. Kashaev. This leads to a slightly finer asymptotic
description of the invariant than predicted by the volume conjecture of Kashaev
and H. Murakami and J. Murakami.
