$(1)$ Determine the order of the following expressions as $\epsilon \rightarrow 0$ $$(i) \space ln [ 1+\frac{1+2\epsilon}{\epsilon(1-2\epsilon)} ]$$ $$(ii) \space \frac{1-cos \epsilon}{1+cos\epsilon}$$
My attempts:
$(1i) => ln[1+\frac{1+2\epsilon}{\epsilon - 2 \epsilon^2}]$ $=>\epsilon - 2\epsilon^2 \approx \epsilon$ $=>O(ln(\epsilon))$ or $O(\epsilon)$
$(1ii) =>$ no order? Since as $\epsilon \rightarrow 0$ $=>\lim_{\epsilon \rightarrow 0} \frac{1-cos \epsilon}{1+cos\epsilon} = \frac{1-1}{1+1} = 0$