Supporse that $$ f(x)=\frac{1}{|x|^\alpha+1},\quad\alpha\in(0,1],x\in I:=[-1,1], $$ and $$ g(x,y)=\frac{|f(x)-f(y)|}{|x-y|^\beta},\quad x,y\in I. $$ I can prove if $$ \sup_{x,y\in I}g(x,y)<+\infty, $$ then one must require that $\alpha=\beta$.
My question is how to get the exact value of $\sup_{x,y\in I}g(x,y)$ in this case?
The first thing is that $$ \lim_{\substack{x\to0 \\y\to 0}}g(x,y)=1, $$ holds?