induced metric, define $\lim^{\omega}(Y_n)_e$ to be the subset of $Con^{\omega}(X,e,l)$ consisting of $x\in Con^{\omega}(X,e,l)$ that can be represented by a sequence $\{x_n\}$, where $x_n\in Y_n$.

The asymptotic cone is a complete metric space. Moreover asymptotic cone $Con^{\omega}(X,e,l)$ is a geodesic metric space if $X$ is a geodesic metric space [@DS05; @Gr96].

The asymptotic cone of a finitely generated group $G$ with a word metric is asymptotic cone of it’s Cayley graph. Asymptotic cone of a finitely generated group with respect to two generating sets are bi-Lipscitz.\
A geodesic path $p$ in $Con^{\omega}(X,e,l)$ is called limit geodesic if $p=\lim^{\omega}(p_n([0,\lambda_n]))_e$ where $p_n:[0,\lambda_n]\rightarrow X$ is geodesic for all $n\in \mathbb{N}$.

[@DS05] Let $X$ be a complete geodesic metric space. Let $\mathcal{P}$ be a collection of closed geodesic non-empty subsets (called $''pieces''$). If the following properties are satisfied:\
$(T_1)$ Every two different pieces have at-most one common point.\
$(T_2)$ Every nontrivial simple geodesic triangle (A simple loop consists with 3 geodesics) in $X$ is contained in one piece.\
Then we say $X$ is tree-graded space with respect to $\mathcal{P}$.

$(T_2')$ Every non trivial simple loop in $X$ is contained in one piece.\
Here one can replace condition ($T_2$) by ($T_2'$).\
Here by loop we 