Expected Value of a Infinite Process Assume $x_t$ is an unit root process. Write first difference of $x_t$  as

$\Delta x_t$ =$\rho$ $\Delta x_{t-1}$+ $\epsilon_t$

Where $\epsilon_t$ is a white noise process. 

How can we calculate $ E_t [x_{t+j}]$ when $j \rightarrow \infty$.

Suppose:

$a_t$= $x_t$ +$z_t$ 

$\Delta x_t$ =$\rho$ $\Delta x_{t-1}$+ $\epsilon_t$

$ z_t$ =$\rho$ $z_{t-1}$+ $\eta_t$

How can we calculate $ E_t [a_{t+j}]$ when $j \rightarrow \infty$.
You are asking about the expected value of the leading term of a sequence. There can be no general answer given the assumptions you make, since for example

$$x_{t+1} = x_{t} + v_{t+1}, \;\;\;E(v_{t+1}) = 0,\;\;\; x_0 \;\; \text{given}$$

is a unit root process, and its expected value is equal to $E(x_{t+1})=x_0,\;\;\; \forall t$, while

$$x_{t+1} = a +x_{t} + v_{t+1}, \;\;\;E(v_{t+1}) = 0,\;\;\; x_0 \;\; \text{given}$$

is also a unit-root process, but $E(x_{t+1}) = x_0 + a\cdot(t+1)$

which is very different compared to the previous case.