Im trying to solve a problem from renovation theory that says that if a machine gets swaped when it reaches its end of life $X \sim F$ or when a time S has passed then the rate of renovation when t->$\infty$ is: $\frac{F(S)}{\int_{0}^{S}xf(x)dx + S(1-F(S))}$
I know i need to use the theorem of renovation with the limit but im having a hard time with the rest. I tried defining T=min{X,S} and then getting that random variable's distribution but Im not sure if its the right path.