Question

Show that the function $f$ fiven below by:
$\{\begin{array}{cc}\frac{e^{1/x}-1}{e^{1/x}+1}& ,ifx\ne 0\\ -1& ,ifx=0\end{array}$ is discontinuous at $x=0$

Answer 1

${lim}_{h\to 0}f(x+h)=\frac{e^{\frac{1}{h}}-1}{e^{\frac{1}{h}}+1}\Rightarrow \frac{1-e^{\frac{-1}{h}}}{1+e^{\frac{-1}{h}}}\left({lim}_{h\to 0}\right)\Rightarrow \frac{1-0}{1+0}=1(\ne 1){lim}_{x\to 0}f\left(x\right)\ne f\left(x\right)at0hencediscontinuousat0.$