In this paper, we study the extended limit set $J_{\{T_n\}}(x)$ for a sequence of bounded linear operators $\{T_n\}_{n=1}^{\infty}$ on a separable Banach space $X$, to describe the dynamics of a sequence of linear operators. First, it is shown that a mutually commuting sequence $\{T_n\}_{n=1}^{\infty} \subseteq \mathcal{B}(X)$ is hypercyclic if and only if the closed and connected set $H=\{x\in X: J_{\{T_n\}}(x)=X\}$ is dense in $X$ or equivalently has nonempty interior. % Later, we show that a sequence % $\{T_n\}_{n=1}^{\infty}$ with $T_{m+n}=T_mT_n$ % is hypercyclic if and only if there exists a cyclic vector %$x\in X$ such that $J_{\{T_n\}}(x)=X$. Furthermore, for such a %sequence we prove that it is $J$-class if and only if %$J_{\{T_n\}}(0)=X$ or equivalently, $J_{\{T_n\}}(x)=X$ for some % nonzero periodic vector $x$. Later, we prove that a mutually commuting sequence $\{T_n\}_{n=1}^{\infty}$ with dense ranges, is hypercyclic if there exists a cyclic vector $x\in X$ for some term of $\{T_n\}_{n=1}^{\infty}$ such that $J_{\{T_n\}}(x)=X$. Furthermore, it is proved that an arbitrary sequence $\{T_n\}_{n=1}^{\infty}$ is $J$-class if and only if $J_{\{T_n\}}(0)=X$ or equivalently, $J_{\{T_n\}}(x)=X$ for some nonzero mixed periodic vector $x$. Finally, some examples are presented to illustrate these results.
