Why do I first need to bring $-4x$ into the numerator in $\lim_{x\to \infty} 4x^2/(x-2) - 4x$ I tried solving the question in the title as follows:
$$\lim_{x\to \infty} \frac{4x^2}{x-2} - 4x \to 4x - 4x \to 0$$
However, apparently that first step ($\to 4x - 4x$) was wrong, and I should first have brought the second $4x$ into the numerator.
My question is not how I need to solve the question, as I know that now. My question is why what I did was wrong, as I lack any intuition for it, and it seems a mystery to me.

As $x \to \infty$, $\frac{4x^2}{x-2}$ and $4x$ are asymptotically equivalent. However, the notion of asymptotic equivalence is of relative equivalence (in the sense that their ratios tend to $1$). We cannot deduce anything about the differences. The differences may be fixed (e.g. $x^2 \sim x^2 + 1$), tend to $0$ (e.g. $x^2 \sim x^2 + \frac1x$) or tend to infinity (e.g. $x^2  \sim x^2 + x$). 
Therefore when working with limits which involve differences, the asymptotic equivalence becomes essentially irrelevant. This is why what you're doing is incorrect. 
