Verify $\lim _{h \rightarrow 0 } \frac{P(X_{t+h} - X_t \geq 2)}{P(X_{t+h} - X_t = 1)} = 0$

Question

If an inhomogeneous Poisson process has the following properties:

1. $P(X_{t+h} - X_t \geq 2) = o(h) $

2. $P(X_{t+h} - X_t \geq 1) = \lambda (t) h + o(h) $

Use this results to verify:

$$\lim _{h \rightarrow 0 } \frac{P(X_{t+h} - X_t \geq 2)}{P(X_{t+h} - X_t = 1)} = 0$$

I don't know how to work with the fraction so that I end up with $\lim _{h \rightarrow 0 } \frac{o(h)}{h} $ which by definition is $0$. Thanks for the help.

Answer 1

$\frac {o(h)} {\lambda (t) h+o(h)}=\frac {o(1)} {\lambda (t)+o(1)}$ after division by $h$. Hence, the limit is $\frac 0 {\lambda (t)+0}=0$.