مشتق چپ تابع $f(x)=\left\{ \begin{matrix} \sqrt{۱+x-۲\sqrt{x}}\,\,\,;\,\,\,x\notin Z \\ ۰\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,x\in Z \\ \end{matrix} \right.$ در نقطهی $x=۱$ کدام است؟
${{{f}'}_{-}}=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{f(x)-f(1)}{x-1}=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{\sqrt{1+x-2\sqrt{x}}-0}{x-1}=\frac{0}{0}$ $\Rightarrow \underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{\sqrt{{{(\sqrt{x}-1)}^{2}}}}{x-1}=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{\left( \sqrt{x}-1 \right)}{(\sqrt{x}-1)(\sqrt{x}+1)}=\frac{-1}{2}$