This paper is concerned with asymptotic analysis of positive decreasing solutions of the secondorder quasilinear ordinary differential equation \begin{equation}ag{E} (p(t)ǎrphi(|x'(t)|))'=q(t)si(x(t)), \end{equation} with the regularly varying coefficients $p$, $q$, $\varphi$, $\psi$. An application of the theory of regular variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that $\lim\limits_{t\to\infty}x(t)=0$, $\lim\limits_{t\to\infty}p(t)\varphi(-x'(t))=\infty$.