I am trying to understand the statement $$ \frac{1}{1-x} = 1 + \Theta(x) $$ for $0 < x < 1$.
To my understanding, this could mean two things:
There are constants $C_1$ and $C_2$ such that $1+C_1 x \leq \frac{1}{1-x} \leq 1 + C_2 x$ for all $x \in (0,1)$. However, this is false as such a $C_2$ does not exist.
There exist constants $C_1$ and $C_2$ such that $1 + C_1x \leq \frac{1}{1-x} \leq 1 + C_2x$ for all large (or small) enough $x$. However, this is not a statement about $x$, so in that case I cannot make sense of the "for $0<x<1$" part.
So how should this statement be interpreted? Any help is appreciated.