  We present a systematic study for the limit closure $(\underline{x})^{\lim}$
of a sequence of elements $\underline{x}$ (eg. a system of of parameters) in a
local ring. Firstly, we answer the question which elements are always contained
in the limit closure of a system of parameters. Then we apply this result to
give a characterization of systems of parameters which is a generalization of
previous results of Dutta and Roberts in \cite{DR} and of Fouli and Huneke in
\cite{FH}. We also prove a topological characterization of unmixed local rings.
In two dimensional case, we compute explicitly the limit closure of a system of
parameters. Some interesting examples are given.
