Is the following statement true?
Assume that $(M^n, g, p)$ is a pointed complete manifold with metric $g$, $Rc(g)\geq 0$, 
$\{s\_i\}$ is a positive sequence decreasing to $0$, and $(M^n, s\_i\cdot g, p)$ converges to metric space $(N, h, p)$, where $h$ is the metric on $N$. Then we always have that 
$h= dr^2+ f(r)^2 dX$, where $r$ is the distance function from point $p$, $X$ is another compact metric space.



the tangent cone at infinity of manifolds with Rc\geq 0
The answer is no. There are manifold with positive Ricci curvature and tangent cone at infinity is not polar. The example is constructed by Menguy "Examples of nonpolar limit spaces. Amer. J. Math. 122 (2000), no. 5, 927–937". 


