14. As shown in the figure, given a geometric sequence $\left\{a_{n}\right\}$ with all terms positive, the total number of terms is 1, the common ratio is $x$, the sum of the first $n$ terms is $S_{n}$, and set $f(x)=\lim _{n \rightarrow \infty} \frac{2 S_{n}}{S_{n+1}}$.
(1) Find the analytical expression for $f(x)$;
(2) Sketch the graph of the function $y=f(x)$.
14. (1) When the common ratio $x=1$, $f(x)=\lim _{n \rightarrow+\infty} \frac{2 n}{n+1}=2$. When $x>1$, $f(x)=\lim _{n \rightarrow+\infty} \frac{2\left(1-x^{n}\right)}{\left(1-x^{n}\right)+(1-x)}=2$; When $0<x<1$, $f(x)=\lim _{n \rightarrow+\infty} \frac{2\left(1-x^{n}\right)}{\left(1-x^{n}\right)+1-x}=\frac{2}{2-x}$. Therefore, $f(x)=\left\{\begin{array}{l}2, x \geqslant 1 ; \\ \frac{2}{2-x}, 0<x<1 ;\end{array}\right.$
(2) Graph omitted.