This paper is concerned with the asymptotic behavior of solutions of the Volterra integral equation \[x(t) + \int_0^t {a(t,\tau )g(x(\tau ))d\tau = f(t)} ,\quad 0 \leqq t < \infty \] If $x(t)$ is a solution of this equation, the limiting values of $g(x(t))$ are given under various sets of hypotheses on the kernel $a(t,\tau )$ and the functions $g(t)$ and $f(t)$.