Let $f(x)$ be continuous for $x>0$; we consider the existence and asymptotic behavior for $c\to\infty$ of nonoscillatory solutions of the equation \[ y''+f(x)y=0 \] In our analysis there are neither hypotheses concerning the sign of the function $f(x)$ nor hypotheses concerning the absolute integrability of $f(x)$ over $(\alpha,\infty)$. First we prove a sufficient condition for the existence of nonoscillatory solutions of (1.1) and then obtain asymptotic formulae for such solutions.