If $\displaystyle f:X \rightarrow \mathbb{R}$ is defined by $\displaystyle f(x)=\frac{x}{x^2-1}$, and we are interested in the behavior of $\displaystyle f$ near $\displaystyle 0$, then $\displaystyle 1,-1 \not\in X$. A natural choice of $\displaystyle X$ is $\displaystyle \mathbb{R}-\{-1,1\}$, in which case $\displaystyle (-\delta, \delta) \subseteq X$ for every real number $\displaystyle \delta$ such that $\displaystyle 0<\delta \leq 1$.

It's seems in every calculus book I read that the choice of $\displaystyle \delta$ has always been $\displaystyle 0<\delta \leq 1$.

Could someone explain why $\displaystyle 0<\delta \leq 1$?