  In this paper, we consider one--parameter ($\lambda>0$) families of Li\'enard
differential equations. We are concerned with the study on the asymptotic
behavior of periodic solutions for small and large values of $\lambda>0$. To
prove our main result we use the relaxation oscillation theory and a
topological version of the averaging theory. More specifically, the first one
is appropriate for studying the periodic solutions for large values of
$\lambda$ and the second one for small values of $\lambda$. In particular, our
hypotheses allow us to establish a link between these two theories.
