Inverse unit compound Rayleigh distribution: statistical properties with applications in different fields

$$\begin{aligned} \underset{x \rightarrow 1}{\lim }\ f(x)= 2\tau \rho ^\tau \underset{x \rightarrow 1}{\lim }\ \left[ (x-1)^{2\tau -1}\left( \rho (x-1)^2+1\right) ^{-(\tau +1)}\right]…