  Using Hitchin's parameterization of the Hitchin-Teichm\"uller component of
the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of
certain families of representations. In fact, for certain Higgs bundles in the
$SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian
metric solving the Higgs bundle equations. This analysis is used to estimate
the asymptotics of the corresponding family of flat connections as we scale the
differentials by a real parameter. We consider Higgs fields that have only one
holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We
also study the asymptotics of the associated family of equivariant harmonic
maps to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ and relate it
to recent work of Katzarkov, Noll, Pandit and Simpson.
