  We study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary
invariant ensembles with Freud-type and uniform convex potentials. As both $k$
and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the
bulk and soft edge cases, respectively. Multi-dimensional central limit
theorems, as well as moderate deviations, are also proved. This work
generalizes earlier results in the GUE and unitary invariant ensembles with
monomial potentials of even degree. In particular, we obtain the precise
asymptotics of corresponding Christoffel-Darboux kernels as well.
