I'd like to examine the continuity of the following function:
f[x_] := Limit[(n^x - n^(-x) ) / (n^x + n^(-x) ), n -> Infinity]
I know that this function isn't continuous because $f(x)=-1$ for $x<0$, for $x=0$ $f(x)=0$, and finally $f(x)=1$ for $x>0$ .
$$ x<0\quad \quad f(x)=-1 \\ x=0\quad \quad f(x)=\ \ \ 0 \\ x>0\quad \quad f(x)=\ \ \ 1 \\ $$
I would rather have a more formal proof than my guess. Could someone give me a tip on how I can prove this using Mathematica?