In my text book, they state the following:
$\begin{align*}f(x) &= (\frac{1}{x} + \frac{1}{2}) (x-\frac{1}{2}x^2+\frac{1}{3}x^3+O(x^4))-1& ,x \rightarrow 0\\&= 1-\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x-\frac{1}{4}x^3+O(x^3)-1& ,x \rightarrow 0 \end{align*}$
However, when I calculate this, I get $1-\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x-\frac{1}{4}x^3+O(x^3)+\frac{O(x^4)}{2}-1$. That $O(x^4)$ part disappears I guess, due to the big O notation. However, I cannot figure out why.
Furthermore, a few pages later, they say that $\lim_{x\rightarrow 0} O(x) = 0$. Which I do not really understand, since $O(x)$ defines a set of functions, no?