  Consider a diagram $\cdots \to F_3 \to F_2\to F_1$ of algebraic systems,
where $F_n$ denotes the free object on $n$ generators and the connecting maps
send the extra generator to some distinguished trivial element. We prove that
(a) if the $F_i$ are free associative algebras over a fixed field then the
limit in the category of graded algebras is again free on a set of homogeneous
generators; (b) on the other hand, the limit in the category of associative
(ungraded) algebras is a free formal power series algebra on a set of
homogeneous elements, and (c) if the $F_i$ are free Lie algebras then the limit
in the category of graded Lie algebras is again free.
