  In this paper we consider a scalar field system with a class of potentials
given by the expression, $V(\phi)\propto \phi^m {\rm exp}({-\lambda
\phi^n/{M^n_{Pl}}})$; $m\geqslant 0, n>1$ for which $\Gamma=V_{\phi
\phi}V/V^2_{\phi}\to 1 $ as $|\phi|\to \infty$. We carry out dynamical analysis
for the underlying system choosing a suitable set of autonomous variables and
find all the fixed points. In particular, we show that the scaling solution is
an attractor of the system in the asymptotic regime. We indicate the
application of the solution to models of quintessential inflation.
