If we are given an infinite system of abelian groups $(\dots \xrightarrow{\varphi_2} A_2 \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ then I know its inverse limit can be found by $$\lim_{i\geq 0} A_i = \ker(id - \varphi):\prod_{i\geq 0}A_i \to \prod_{i\geq 0}A_i $$ where $(id - \varphi)(a_0,a_1,a_2,\dots) = (a_0 - \varphi_0(a_1),a_1 - \varphi_1(a_2),\dots)$.
Can this also be applied to finite sequences, in particular can the limit of $(A_n \xrightarrow{\varphi_{n-1}} \dots \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ be calculated in the same way? As an example I am trying to find the limit of $\mathbb{Z}/8\mathbb{Z} \xrightarrow{\cdot 3} \mathbb{Z}/8\mathbb{Z}$ (which should be $\mathbb{Z}/2\mathbb{Z}$).