定义新运算$m*n=\dfrac{mn+1}{m+n}$，则$\left(\cdots \left(\left(100*99\right)*98\right)*\cdots *3\right)*2$的值是_______．
分析与解 $\dfrac{5051}{5049}$．
注意运算形式，有$$m*n+1=\dfrac{(m+1)(n+1)}{m+n},m*n-1=\dfrac{(m-1)(n-1)}{m+n},$$于是$$\dfrac{m*n+1}{m*n-1}=\dfrac{m+1}{m-1}\cdot\dfrac{n+1}{n-1},$$这就意味着若设$f(x)=\dfrac{x+1}{x-1}$，则有$$f(m*n)=f(m)\cdot f(n).$$令所求式为$A$，则$$f(A)=f(100)\cdot f(99)\cdots f(2)=\dfrac{101!}{99!\cdot 2!}=5050,$$于是可解得$A=\dfrac{5051}{5049}$．