Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.
1. **Identify the given figure and its properties:**
   The given figure is an ellipse described by the equation:
   \[
   F: x^2 + \frac{y^2}{3} = 1
   \]
   This is an ellipse centered at the origin with semi-major axis \(a = 1\) and semi-minor axis \(b = \sqrt{3}\).

2. **Understand the rotation of the ellipse:**
   The ellipse is rotated counterclockwise about the origin by angles \(\frac{k}{2n}\pi\) for \(k = 0, 1, 2, \ldots, n\). This means there are \(n+1\) ellipses, each rotated by a different angle.

3. **Determine the common area of the rotated ellipses:**
   As \(n\) increases, the number of rotated ellipses increases, and they become more densely packed around the origin. The common area of these ellipses will be the region that remains invariant under all these rotations.

4. **Analyze the limiting behavior as \(n \to \infty\):**
   As \(n\) approaches infinity, the ellipses will be rotated by every possible angle between \(0\) and \(\pi\). The common area of all these rotated ellipses will be the largest circle that can fit inside the original ellipse and remain invariant under rotation.

5. **Find the radius of the invariant circle:**
   The largest circle that can fit inside the ellipse \(x^2 + \frac{y^2}{3} = 1\) has a radius equal to the semi-major axis of the ellipse, which is 1. This is because the circle must fit within the bounds of the ellipse in all directions.

6. **Calculate the area of the invariant circle:**
   The area of a circle with radius 1 is given by:
   \[
   \text{Area} = \pi \times (1)^2 = \pi
   \]

7. **Conclude the limit of the common area \(S_n\):**
   Therefore, the limit of the common area \(S_n\) as \(n\) approaches infinity is \(\pi\).

\[
\lim_{n \to \infty} S_n = \pi
\]

The final answer is \(\boxed{\pi}\).