Let $k$ be a field and $A$ a finite dimensional $k$-algebra. Given a sequence of inclusions $M_1 \subseteq M_2 \subseteq \dots$ of $A$-modules consider the direct limit $M:= \bigcup_{i=1}^\infty M_i$. For a finite dimensional module $X$ suppose we have $X\subseteq M$. Then $X$ is generated by finitely many elements $x_1, x_2, \dots, x_n$. Hence we find $m>0$ such that $x_i\in M_m$ for all $m$. It follows that $X\subseteq M_m$. Now consider the dual setting:

Let $M_1 \twoheadleftarrow M_2 \twoheadleftarrow \dots $ be a sequence of $A$-modules, let $M$ be its inverse limit and suppose we find an epimorphism $M\twoheadrightarrow X$, where $X$ is a finite dimensional $A$-module. Is it true in general, that we obtain an epimorphism $M_m \twoheadrightarrow X$ for some $m>0$?