Let $S(t)_{t\geq 0}$ be an integrated semigroup on a vector space $E$. Let $ 0\leq\alpha< 1$, $ x\in E$. I haven't been able to calculate this limit, but I expect it to be zero. $$ \lim_{\epsilon \rightarrow 0^+}\dfrac{S(\epsilon)(x)}{\epsilon^{\alpha}} $$ Does this limit exist?
If $\alpha=0$, this case is easy, $ \lim_{\epsilon \rightarrow 0^+}S(\epsilon)(x)=0 $$
but if $\alpha\neq 0$ what can i do?