  For a hyperbolic knot and a natural number n, we consider the Alexander
polynomial twisted by the n-th symmetric power of a lift of the holonomy. We
establish the asymptotic behavior of these twisted Alexander polynomials
evaluated at unit complex numbers, yielding the volume of the knot exterior.
More generally, we prove the asymptotic behavior for cusped hyperbolic
manifolds of finite volume. The proof relies on results of M\"uller, and
Menal-Ferrer and the last author. Using the uniformity of the convergence, we
also deduce a similar asymptotic result for the Mahler measures of those
polynomials.
