= Solution
When all rates are positive, finite irreducible convergence gives $P(t)\to\mathbf1\pi$, with $\pi$ as above. With one zero-rate state $z$, each visit to either transient state has probability $1/2$ of entering $z$ on the next jump. Survival for $n$ jumps therefore has probability $2^{-n}$, so absorption is almost sure and
$$
\boxed{P_{ij}(t)\to\mathbf1_{\{j=z\}}.}
$$
With two zero-rate states $z_1,z_2$ and the sole positive-rate state $k$, absorption occurs at the first jump, with probability $1/2$ into each absorbing state. The limit rows are $e_{z_1}^T,e_{z_2}^T$ for those states and $(e_{z_1}^T+e_{z_2}^T)/2$ for row $k$. These limits depend on the initial state, as nonuniqueness requires.
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