$\text{ Let }f\left(x\right)=\left\{\begin{array}{l}\frac{x}{1+2^{1/x}},\text{ if }x\ne 0\\ 0\text{ if }x=0\end{array}\text{ then }{f}^1\left(0^{-}\right)=\right.$

$\begin{array}{l}{f}^1\left(0^{-}\right)=\underset{x\to 0^{-}}{lim} \frac{f\left(x\right)-f\left(0\right)}{x-0}\\ =\underset{x\to 0^{-}}{lim} \frac{1+2^{\frac{1}{x}}}{x}\\ =\frac{1}{1+2^{-\frac{1}{0}}}\\ =\frac{1}{1+2^{-\mathrm{\infty }}}\\ =\frac{1}{1+0}\\ =1\end{array}$