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A criterion is given for the equation — (py')' + qy = 0 to have a solution on the interval [α, °°) which is not in L2(a,o°). The criterion permits q (or Reg if q is complexvalued) to be decomposable into a sum q = qλ + q2 where the expression ~ (py')' + qiy essentially satisfies the well-known limit-point criterion of Levinson and q2 may be thought of as an oscillatory function whose amplitude may be large, but whose integral over [a, x] increases relatively slowly as a function of JC.

A limit-point criterion for expressions with oscillatory coefficients