Torus Limit

Consider all curves $f_{m}\left(x,y\right)$ of the form $x^2+(y-m)^2=m^2$ for $m\in\mathbb{R}$ , and define the following:

• Let $V_n$ denote the average value from $m=-n$ to $m=n$ of the volume when $f_{m}\left(x,y\right)$ is rotated about the $x$ -axis.
• Let $S_n$ denote the average value from $m=-n$ to $m=n$ of the surface area when $f_{m}\left(x,y\right)$ is rotated about the $x$ -axis.

Evaluate $\lim\limits_{n\to\infty}\dfrac{V_n}{nS_n}$ .