This is to supplement Otis Chodosh's fine answer. From Robert Bryant's analysis of his steady Ricci soliton, one has
$g=dr^{2}+\phi(r)^{2}g_{S^{n-1}}$, where
$$
C^{-1}r^{1/2}\leq\phi(r)\leq Cr^{1/2},\quad\phi^{\prime}(r)=O(r^{-1/2}
),\quad\phi^{\prime\prime}\left( r\right) =O(r^{-3/2}).
$$
Consider a basepoint $x_{i}$ on the $(n-1)$-sphere with $\{r(x)=r_{i}\}$ and
rescale so that this sphere has radius $1$; i.e., let
$$
g_{r_{i}}=\frac{dr^{2}}{\phi(r_{i})^{2}}+\left( \frac{\phi(r)}{\phi(r_{i}
)}\right) ^{2}g_{S^{n-1}}=ds^{2}+\left( \frac{\phi(\phi(r_{i})s+r_{i})}
{\phi(r_{i})}\right) ^{2}g_{S^{n-1}},
$$
where $s=\phi(r_{i})^{-1}(r-r_{i})$. We have
$$
\frac{d}{ds}\left( \frac{\phi(\phi(r_{i})s+r_{i})}{\phi(r_{i})}\right)
=\phi^{\prime}(\phi(r_{i})s+r_{i})=O((\phi(r_{i})s+r_{i})^{-1/2}
)=O(r_{i}^{-1/2}),
$$
provided $s=O(\phi(r_{i})^{-1}r_{i})=O(r_{i}^{1/2})$. Now choose any sequence
$\{x_{i}\}$ so that $r_{i}=r(x_{i})\rightarrow\infty$. From this one obtains a
pointed cylinder limit of $(g_{r_{i}},x_i)$ in the Cheeger-Gromov sense without having to pass to a subsequence.

In some more general situations, one can apply Hamilton's Cheeger-Gromov type
compactness theorem for complete solutions to the Ricci flow with semi-global
curvature bounds. Some form of dimension reduction (given a line or using the
strong maximum principle) together with Hamilton's classification (reproved by
Perelman) of $2$-dimensional $\kappa$-solutions is also useful here.

As an analogue (although extrinsic), consider a parabola $y=x^{2}$ in the plane. Dimension reduction
of this curve yields two parallel lines as follows. Let $x_{i}>0$ and consider
the basepoint $(x_{i},x_{i}^{2})$. Multiply the parabola by the constant
$x_{i}^{-1}$ so that distance from $(x_{i},x_{i}^{2})$ to the $y$-axis becomes
$1$. We get $x_{i}y=(x_{i}x)^{2}$, i.e., $y=x_{i}x^{2}$. Now translate this
curve so that $(1,x_{i})$ goes to $(1,0)$. The curve becomes $y=x_{i}
x^{2}-x_{i}$. As $x_{i}\rightarrow\infty$ this curve converges in compact
subsets of the plane to the pair of vertical lines $x=\pm1$, i.e.,
$S^{0}\times\mathbb{R}$.