  In this paper we use motivic integration and non-archimedean analytic
geometry to study the singularities at infinity of the fibers of a polynomial
map $f\colon \mathbb A^d_\mathbb C \to \mathbb A^1_\mathbb C$. We show that the
motive $S_{f,a}^{\infty}$ of the motivic nearby cycles at infinity of $f$ for a
value $a$ is a motivic generalization of the classical invariant
$\lambda_f(a)$, an integer that measures a lack of equisingularity at infinity
in the fiber $f^{-1}(a)$. We then introduce a non-archimedean analytic nearby
fiber at infinity $\mathcal F_{f,a}^{\infty}$ whose motivic volume recovers the
motive $S_{f,a}^{\infty}$. With each of $S_{f,a}^{\infty}$ and $\mathcal
F_{f,a}^{\infty}$ can be naturally associated a bifurcation set; we show that
the first one always contains the second one, and that both contain the
classical topological bifurcation set of $f$ whenever $f$ has isolated
singularities at infinity.
