What is the resulting big oh for this expression? I'm new to big-oh notation and working with its arithmetic. If I'm given the following
$a=O(\delta^2),\quad b=O(\delta^2),\quad c=O(\delta^2),\quad d=O(\alpha^2)$
What is the the resulting big-oh for the following expression if $(\alpha \rightarrow 0,\delta \rightarrow 0)$? For the sake of further understanding, how about $(\alpha \rightarrow \infty,\delta \rightarrow \infty )$?
$$\displaystyle \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$
I understand there are wikipedia pages and other sources explaining the arithmetic, but I can not follow. I need to learn by example. Please show how the powers, products, etc. influence the orders. I will later have to change the orders of $a,b,c,d$ and I need to know the correct way to find the resulting big-oh of the expression. For example, I have to also find the result of
$a=O(\delta^4),\quad b=O(\delta^4),\quad c=O(\delta^2),\quad d=O(\alpha^2)$
But again, if you can show by example with the expression I've given and highlight the arithmetic, I should be fine moving forward.
EDIT: I made changes based on the comments. I've also changed $d=O(\delta^2)$ to $d=O(\alpha^2)$ since it depends on another variable. The root of this question comes from expressions for the truncation error. For example the variable $a$ is expressed as a truncated Taylor series to $O(\delta^2)$. All variables are truncated series expressions, but I don't understand how the truncation of each variable effects the final truncation of the expression. Please let me know if there is more I need to clarify.

You can't answer this question, because the ratio of two functions known by an upper bound only (Big-$O$) is indeterminate.
For instance, both $\delta=O(\delta^2)$ and $\frac1{\delta^{27}}=O(\delta^2)$ hold, while $$\dfrac\delta{\frac1{\delta^{27}}}=\delta^{28}.$$
Actually, Big-$\Theta$ is required.
