= Solution
State one is left after one step and never revisited. The eventual probability mass in the closed class $\{4,5\}$ is
$$
w=(1-p)\lambda_1+\lambda_4+\lambda_5.
$$
Within this class the <stationary distribution> is $(b/(a+b),a/(a+b))$, because the balance equation is $a\pi_4=b\pi_5$. The two-state recurrence shows convergence from either starting state, since $|1-a-b|<1$. The other closed class never reaches state five. Therefore
$$
\boxed{\lim_{n\to\infty}\mathbb P(X_n=5)
=\frac a{a+b}\bigl((1-p)\lambda_1+\lambda_4+\lambda_5\bigr)}.
$$
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