In general direct and inverse limits do not commute but in some cases it does (Ref: Handbook of Categorical Algebra, Volume 1: Basic Category Theory by Francis Borceux Chapter 2). The question here concerns a specific example.
For concreteness let us work over commutative rings, so that inverse limit $\varprojlim$ preserves injection (but not surjection) and direct limit $\varinjlim$ is an exact functor. Let $A_i$ be complete (say with respect to some ideal) and thus (by assumption) $\varprojlim_\lambda A_i=A_i$. Consider the following directed system $$ \begin{equation}\label{one} A_1\hookrightarrow A_2\hookrightarrow \ldots\hookrightarrow A_i\hookrightarrow\ldots\hookrightarrow \varinjlim_i A_i=\bigcup_i A_i \end{equation} $$
Apply inverse limit to the above and since by assumption $\varprojlim A_i=A_i$ this gives $$ A_1\hookrightarrow A_2\hookrightarrow \ldots\hookrightarrow A_i\hookrightarrow\ldots\hookrightarrow \varprojlim_\lambda\varinjlim_i A_i $$
The universal property of direct limit gives the map $\varinjlim_i A_i\rightarrow\varprojlim_\lambda\varinjlim_i A_i$ and universal property of inverse limit gives the map $\varprojlim_\lambda\varinjlim_i A_i\rightarrow\varinjlim_i A_i $.
Question Can we conclude that in the above case $\varinjlim_i A_i=\varprojlim_\lambda\varinjlim_i A_i$? Why?
Conditional Example If above holds it gives an example of $\varprojlim_\lambda\varinjlim_i A_i=\varinjlim_i\varprojlim_\lambda A_i$.