Show that `lim_(xto0) (e^(1//x)-1)/(e^(1//x)+1)` does not exist.

Explanation

Let `f(x)=(e^(1//x)-1)/(e^(1//x)+1)`
L.H.L of `f(x) " at "x=0` is
`underset(xto0^(-))limf(x)=underset(hto0)lim(0-h)=underset(hto0)lim(e^(-1//h)-1)/(e^(-1//h)+1)`
`underset(hto0^(-))lim(((1)/(e^(1//h))-1)/((1)/(e^(1//h))+1))=-1`
`[becausehto0implies1/htoooimpliese^(1//h)toooimplies(1)/(e^(1//h))to0]`
R.H.L. of `f(x)` at `x=0` is
`underset(xto0)limf(x)=underset(hto0)limf(0+h)=underset(hto0)lim(e^(1//h)-1)/(e^(1//h)+1)`
`=underset(hto0)lim((1-(1)/(e^(1//h)))/(1+(1)/(e^(h))))" "`[Dividing Nr and Dr by `e^(1//h`)]
`=(1-0)/(1+0)=1`
Clearly, `underset(xto0^(-))limf(x)neunderset(xto0^(+))limf(x )`
Hence, `underset(xto0)limf(x)` does not exist.