Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / \varepsilon \rceil} = \underset{\varepsilon \rightarrow 0}{lim} \hspace{3pt} e^{\frac{log(1 - p_\varepsilon)}{\varepsilon}t} \in [0, 1]. $$ I don't understand the above relation. $$e^{\frac{log(1 - p_\varepsilon)}{\varepsilon}t} = e^{\frac{t}{\varepsilon} log(1-p_\varepsilon)} $$ and then?
Thank you!