It is proved that LIM (1-e ^ 1 / x) / (1 + e ^ 1 / x) does not exist when x tends to 0 LIM (1 / X of 1-e) / (1 / X of 1 + e) does not exist when x tends to zero

It is proved that LIM (1-e ^ 1 / x) / (1 + e ^ 1 / x) does not exist when x tends to 0 LIM (1 / X of 1-e) / (1 / X of 1 + e) does not exist when x tends to zero

The original formula = LIM (x - > 0) {[2-1-e ^ (1 / x)] / [1 + e ^ (1 / x)]}
=lim(x->0){2/[1+e^(1/x)]-1}
∵ right limit = LIM (x - > 0 +) {2 / [1 + e ^ (1 / x)] - 1} = - 1
Left limit = LIM (x - > 0 -) {2 / [1 + e ^ (1 / x)] - 1} = 1
‖ right limit ≠ left limit
So LIM (x - > 0) (1-e ^ 1 / x) / (1 + e ^ 1 / x) = does not exist