Question

$f\left(x\right)=\frac{2^{1/x}}{1+2^{1/x}},x\ne 0$ and $f\left(0\right)=1$ Is function continuous at x=0 ?

Answer 1

Rearranging f(x)

$f\left(x\right)=\frac{1}{2^{-1/x}+1},x\ne 0$

$f(0+0)=\stackrel{Lt}{h\to 0}\frac{1}{2^{-1/h}+1}=\frac{1}{0+1}=1$

$f(0-0)=\stackrel{Lt}{h\to 0}\frac{1}{2^{1/h}+1}=\frac{1}{\infty +1}=0\ne f(0+0)$

Hence disontinuous at $x=0.$