birth-death process

Question

This is from notes

I don't understand why $e^{-h(\lambda _i+\mu _i)}=1$?

and then the note shows that:

But isn't $o(h)$ the probility of combinations such as 2 births and 1 death, 3 births and 2 deaths, etc, so $|X(t+h)-X(t)|$ is still 1, why the note says it is $P|X(t+h)-X(t)|>1$

Answer 1

It isn't $1$, it is $1+o(h)$.

The $o(h)$ term is combined with other $o(h)$ terms so it looks like it disappears.