  In this paper we first introduce 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 $X$ . Then we study the dynamics of sequence of linear operators
by using the extended limit set. It is shown that the extended limit set is
strongly related to the topologically transitive of a sequence of linear
operators. Finally we show that a sequence of operators
$\{T_n\}_{n=1}^{\infty}\subseteq \mathcal{B}(X)$ is hypercyclic if and only if
there exists a cyclic vector $x\in X$ such that $J_{\{T^n\}}(x)=X$.
