Example 7 There is a sequence of curves $P_{0}, P_{1}, P_{2}, \cdots$. It is known that the figure enclosed by $P_{0}$ is an equilateral triangle with an area of 1, and $P_{k+1}$ is obtained by performing the following operation on $P_{k}$: divide each side of $P_{k}$ into three equal parts, construct an equilateral triangle outward using the middle segment of each side as the base, and then remove the middle segment $(k=0,1,2, \cdots)$. Let $S_{n}$ be the area of the figure enclosed by the curve $P_{n}$.
(1) Find the general term formula for the sequence $\left\{S_{n}\right\}$;
(2) Find $\lim S_{n}$.

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Solution: From the problem, it is easy to see that each side of $P_{0}$ becomes 4 sides of $P_{1}$, so the number of sides of $P_{1}$ is 3 $\times 4$. Similarly, for $P_{1}$, each side of $P_{1}$ becomes 4 sides of $P_{2}$, so the number of sides of $P_{2}$ is $3 \times 4^{2}$. Therefore, the number of sides of $P_{n}$ is $3 \times 4^{n}$.

Given that the area of $P_{0}$, $S_{0}=1$, comparing $P_{1}$ with $P_{0}$, it is easy to see that $P_{1}$ adds a small equilateral triangle to each side of $P_{0}$, with an area of $\frac{1}{3^{2}}$, and $P_{0}$ has 3 sides, so $S_{1}=S_{0}+3 \cdot \frac{1}{3^{2}}=1+\frac{1}{3}$. Comparing $P_{2}$ with $P_{1}$, it is known that $P_{2}$ adds a small equilateral triangle to each side of $P_{1}$, with an area of $\frac{1}{3^{2}} \times \frac{1}{3^{2}}$, and $P_{1}$ has $3 \times 4$ sides, so $S_{2}=S_{1}+3 \times 4 \times \frac{1}{3^{2}}=1+\frac{1}{3}+\frac{4}{3^{3}}$.
Similarly, $S_{3}=S_{2}+3 \times 4^{2} \times \frac{1}{3^{6}}=1+\frac{1}{3}+\frac{4}{3^{3}}+\frac{4^{2}}{3^{5}}$.
Conjecture $S_{n}=1+\frac{1}{3}+\frac{4}{3^{3}}+\frac{4^{2}}{3^{5}}+\cdots+\frac{4^{n-1}}{3^{2 n-1}}=\frac{8}{3}-\frac{3}{5} \times\left(\frac{4}{9}\right)^{n}$.
The following is a proof by mathematical induction:
When $n=1$, it is obviously true.
Assume that when $n=k$, $S_{k}=\frac{8}{5}-\frac{3}{5} \times\left(\frac{4}{9}\right)^{k}$.
When $n=k+1$, it is easy to see that after the $(k+1)$-th operation, comparing $P_{k+1}$ with $P_{k}$, $P_{k+1}$ adds a small equilateral triangle to each side of $P_{k}$, with an area of $\frac{1}{3^{2(k+1)}}$, and $P_{k}$ has $3 \times 4^{k}$ sides, so $S_{k-1}=S_{k}+3 \times 4^{k} \times \frac{1}{3^{2(k+1)}}$ $=S_{k}+\frac{4^{k}}{3^{2(k+1)-1}}=S_{k}+\frac{4^{k}}{3^{2 k+1}}=\frac{8}{5}-\frac{3}{5} \times\left(\frac{4}{9}\right)^{k+1}$.
In summary, $S_{n}=\frac{8}{5}-\frac{3}{5} \times\left(\frac{4}{9}\right)^{n}\left(n \in \mathbf{N}^{*}\right)$.
(2) $\lim _{n \rightarrow \infty} S_{n}=\lim _{n \rightarrow \infty}\left[\frac{8}{5}-\frac{3}{5} \times\left(\frac{4}{9}\right)^{n}\right]=\frac{8}{5}$.
Analysis: In many cases, mathematical induction is combined with sequences, which is increasingly common in recent competitions and should be proficiently mastered.