My question is, how to evaluate and prove the situation when we have something like , say $o[\frac1{2x}+o(\frac 1x)]=o[\frac 1x]$.
In which I encountered this case while calculating the limit
$$\lim_{x\to\infty}x\left[\frac 1e-\left(\frac{x}{x+1}\right)^x\right]$$
And as I haven't found a comprehensive question involving the arithmetic of landau symbols, can you offer a generalization (Axiomization) for arithmetics of Big $O$ and small $o$?
Including Composites of them (resp. mixed and single), addition and subtraction of them, exponential of them, and so on. And if you think a book would be suitable for this question, what is it?
