Question

Show that the function f given by:

`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`

is discontinuous at x = 0.

Answer 1

`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`

LHL: `lim_(x → 0^-) (e^(1/x) - 1)/(e^(1/x) + 1)`

= `lim_(h → 0) (e^(-1/h) - 1)/(e^(-1/h) + 1) = (0 - 1)/(0 + 1) = - 1`

RHL: `lim_(h → 0) (e^(1/h) - 1)/(e^(1/h) + 1)`

= `lim_(h → 0) (1 - e^(-1/h))/(1 + e^(-1/h)) = 1`

LHL ≠  RHL
∴ f(x) is discontinuous at x = 0.