I am going over previous mock exams in preparation for an upcoming exam and am having problems with parts (ii) and (iv) and was looking for some guidance.
For part (ii), my thinking was that the limiting probabilities are zero when i does not equal j, since states 1 and 2 are not in the same recurrence classes, and that limiting probability of being in state 2 starting from 2 is 0.5, while the limiting probability of being in state 1 starting from state 1 is 0.4. I found those probabilities using the standard procedure for finding stationary distributions.
For part iv), my thinking was that in the long run, starting from state three, we go to state one half of the time, since the transition probability from 3 to 1 is 0.5. I really don't have an idea how to do this problem though. Any help would be great!
(iv) To visit state $1$ starting from state $3$, one needs that the first step out of state $3$ is to state $1$, not to state $2$. This happens with probability $q=\frac{0.5}{0.5+0.1}$. Assuming that one transitioned to state $1$ rather than to state $2$, one is now forever in the class $C=\{1,4\}$, which is aperiodic, hence the probability to be at state $1$ converges to $\pi(1)$, where $\pi$ is the stationary probability measure of the Markov chain restricted to $C$. According to (ii), $\pi(1)=0.4$ hence $p_{31}(n)\to q\cdot\pi(1)=\frac56\cdot0.4=\frac13$.