  We show that a direct limit of surjections of (weak) Golod--Shafarevich
algebras is a weak Golod--Shafarevich algebra as well. This holds both for
graded and for filtered algebras provided that the filtrations are induced by
the filtration of the first entry of the sequence. It follows that the limit is
an algebra of exponential growth. An example shows that the assumptions of this
theorem cannot be directly weakened.
