This question is a continuation of (Commutativity of Direct and Inverse Limits for a special case.) and (Direct Limit of Power Series)
Let $A_i=\mathbb{Z}_p\langle X_1,\ldots X_i \rangle$ Tate algebra obtained from $p$ adic completion of $\mathbb{Z}_p[X_1,\ldots, X_n]$ and thus (by assumption) $\varprojlim_\lambda A_i=A_i$, where $\varprojlim_\lambda$ is $p$ adic completion. 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 1 Can we conclude that in the above case $\varinjlim_i A_i=\varprojlim_\lambda\varinjlim_i A_i$? Why?
Question 2 If above not true, is $\varprojlim_\lambda\varinjlim_i A_i=\mathbb{Z}_p\langle X_1,X_2,\ldots,X_i,\ldots \rangle$, it is known that $\varinjlim_iA_i=\bigcup_i\mathbb{Z}_p\langle X_1,\ldots, X_i\rangle$.