So I'm trying to show that for $x\rightarrow \infty$:
$(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$
So these complicated big-Oh expressions are clearly going to be a recurring theme in my book, and I simply have no idea how to manipulate them in a rigorous fashion. I know logically what these expressions are saying, for instance this one means that:
For any function $f(x)=O(x^{-1})$ for all $a
Nevertheless my book provides no list of 'legal' operations for big-Oh notation, and no examples of how to work with expressions that don't fall into the most simple category of $f(x) = O(g(x))$. Thus I'm simply at a loss not only for how I go about proving such expressions but I don't even know what operations I'm allowed to perform. How does one generally approach these problems? Thanks.