Let $(P_t)_{t\geq 0}$ a semi group and $f\in D(L)=\left\{f\in \mathcal C_0\mid \lim\limits_{t\to 0}\frac{P_tf-f}{t}\text{ exist}\right\}$ and $\mathcal C_0$ is the set of continuous function s.t. $\lim\limits_{t\to \infty }f(t)=0$.
Set $$Lf=\lim_{t\to 0}\frac{P_t f-f}{t}.$$
Suppose $f\in D(L)$. We have $$\lim_{s\to 0}\frac{P_sP_tf-P_tf}{t}=\lim_{s\to 0}P_t\frac{P_sf-f}{s}=P_tLf.$$
I don't understand the last equality, i.e. why $$\lim_{s\to 0}P_t\frac{P_sf-f}{s}=P_t\lim_{s\to 0}\frac{P_sf-f}{s} \ ?$$