Say that we have some number $x$ that is much less than $y$. My question is, if we have some expression $x + y$, then $x$ could be assumed to be small enough such that the expression is approximately $y$. But, if we square this expression, then $(x + y)^2 = x^2 + y^2 + 2xy$. The $x^2$ will be removed (as it sufficiently small), but what about $xy$? Can it really be kept or should it be assumed to be sufficiently small or neither?
I'm having a bit of a trouble thinking about this and how one can actually approximate this stuff. For example, if $x = 0.01$ and $y = 1$, then $2xy$ will be $0.01$ and it would be assumed to be small enough such that the expression is just $y^2$. But, if we choose diferent values, say $x = 1$ and $y = 100$, then $xy = 100$ which is a very big difference. Then we can no longer say that $(x + y)^2 \approx y^2$. (Perhaps these numbers may not be too large or small to be classified as "much less than" or "much greater than" (how does one really classify that in the first place?)) but it seems that depending on the values, $xy$ can be either kept or not kept.
I want to give an exact example:
If $y$ is large enough and if
$$x=\frac ay, ~y \gg a,~ a\in\mathbb R$$
holds, then we can consider
$$(x+y)^2=\left(\frac ay+y\right)^2≈y^2+2a$$
I think, this example is consistent with $x \ll y$ condition.