Let $T\sim EXP(1/\mu)\implies E(T)=1/\mu\implies f(t)=\mu e^{-\mu t}$
Define $Y:=$ \begin{cases} (1-v^T)/\delta, & \text{if 0 < T < n} \\ (1-v^n)/\delta, & \text{if T $\ge$ $n$ } \end{cases}
where $\delta\in[1,\infty), v\in(0,1),\mu\in(0,\infty)$
Basically, a limit has been put on $Y$.
What is the CDF of $Y$? In other words, what is $P(Y\le y)$?
I don't think that the CDF technique will work here, but I'm not sure. How can you get this CDF?