Free limits of free algebras

Consider a diagram ⋯→F₃→F₂→F₁ 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.