  We analyze the eigenvalue density for the Laguerre and Jacobi
$\beta$-ensembles in the cases that the corresponding exponents are extensive.
In particular, we obtain the asymptotic expansion up to terms $o(1)$, in the
large deviation regime outside the limiting interval of support. As found in
recent studies of the large deviation density for the Gaussian
$\beta$-ensemble, and Laguerre $\beta$-ensemble with fixed exponent, there is a
scaling from this asymptotic expansion to the right tail asymptotics for the
distribution of the largest eigenvalue at the soft edge.
