Conditions are given which imply that a system
X
′
(
t
)
=
F
[
t
,
X
(
g
(
t
)
)
]
X’(t) = F[t,X(g(t))]
has a solution with some components which approach given limits with specified orders of convergence as
t
→
∞
t \to \infty
, while the other components have specified orders of magnitude. The integral smallness conditions on
F
F
permit conditional convergence of some of the improper integrals that occur, and it is not required that
lim
t
→
∞
g
(
t
)
=
∞
{\lim _{t \to \infty }}g(t) = \infty
.