For $x \ll y$, what is the maximum value of $x$ for which it holds true?
I don't expect there to be a definitive answer but I am more so just curious what the general consensus on that is. I usually take $\frac{x}{y} ≤ 0.05$ to be the limit but recently I found a solution to a problem where $\frac{x}{y} = 0.103$ was taken to satisfy $\frac{x}{y} \ll 1$.
It depends on the problem at hand and especially on how accurate you need the solution to be. Often when we say $x \ll y$ we will keep terms proportional to $\frac xy$ but ignore those proportional to $\left( \frac xy \right)^2$. You need the ignored terms not to change the answer too much.