I need a numerical approximation around some functions at $x = 0^{-}$ from the left side, where $x = 0$ is unfortunately the right end of the domain (in the reals) so the functions are not differentiable there.

Nonetheless, all I what is an approximation so I actually just need a one-sided limit along the reals and don't need it to be differentiable. I don't care about the fact that different "series" will be obtained when one approaches $x=0$ from different directions in the complex plane.


That is, I'll take whatever works. 


The two methods shown below of doing series expansion yield results that differ by a complex $i$. There are two examples, each being a **Column** object aligning the two ways of coding that seem mathematically the same to me.

    ClearAll[x];
    Column[{  (* example 1 *)
    Series[ -1 + Sqrt[-x (2 + x)]
     , {x, 0, 4}, Assumptions -> x < 0] 
    , Series[-1 + Sqrt[(2 - x) x]
     , {x, 0, 4}, Assumptions -> x > 0] }]

    Column[{  (* example 2 *)
    Series[  ArcCos[1 + x]
     , {x, 0, 4}, Assumptions ->  x < 0] 
    , Series[ ArcCos[1 - x]
     , {x, 0, 4}, Assumptions -> x > 0] }]

The discrepancy most likely involve something like $\arg(x)$ and this has to do with the expansion being done near the branch cut.

>> However, **why are there two different outcomes in terms of coding?** To me the two ways of coding seem mathematically equivalent. **If one of them is more appropriate, which one?**

The expansion is meant to be, for instance more clearly seen in the 2nd example, $x \to 0^{-}$ coming from the left along the real line (inside the proper domain in reals). In both examples the function goes "vertical" at $x = 0$.

Thank you.