Suppose I have a smooth function $F(x, y, \epsilon, \delta) = \frac{\epsilon f(x)}{\delta y} + g(x) + O(\frac{\epsilon^{2}}{\delta} + \delta)$ where $0< \epsilon, \delta << 1$ are small parameters and $\delta >> \epsilon^{2}$. Here $g$ and $f$ are some smooth bounded functions, expanded in series of $\delta$ and $\epsilon$. If I set $F=0$ and rearrange to get
$$y = -\frac{\epsilon f(x)}{\delta g(x)} + \frac{\epsilon}{\delta} O( \frac{\epsilon^{2}}{\delta} + \delta)$$
Then can I simplify the last term $\frac{\epsilon}{\delta}O( \frac{\epsilon^{2}}{\delta} + \delta)$ ?
If I do so, what do I get? Is it still $O( \frac{\epsilon^{2}}{\delta} + \delta)$ or does it change to $O( \frac{\epsilon^{3}}{\delta^{2}} + \epsilon)$ ?