How can I guarantee the convergence of $f_i(v)$?

Question

Let $f_i:\mathbb{C}^n\rightarrow \mathbb{C}^n$ be a sequence of linear functions and let $v_i$ be a convergent sequence with $\lim v_i=v$. Assume that $f_i(v_i)$ is convergent. What assumptions do I need to impose to guarantee $\lim f_i(v)=\lim f_i(v_i)$? It is clear that if $f_i$ is convergent, then the limit is convergent. I'd like to know if there are other conditions.