Question

Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if"  x ≠ 0),(0",",  "if"  x = 0):}` is discontinuous at x = 0.

Answer 1

The left hand limit of f at x = 0 is given by

`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) ("e"^(1/x) - 1)/("e"^(1/x) + 1)`

= `(0 - 1)/(0 + 1)`

= −1

Similarly, `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) ("e"^(1/x) - 1)/("e"^(1/x) + 1)`

= `lim_(x -> 0^+) (1 - 1/"e"^(1/x))/(1 + 1/"e"^(1/x))`

= `lim_(x -> 0^+) (1 - "e"^((-1)/x))/(1 + "e^((-1)/x)`

= `(1 - 0)/(1 + 0)`

= 1

Thus `lim_(x -> 0^-) "f"(x) ≠  lim "f"(x)_(x -> 0^+)`

Therefore, `lim_(x -> 0) "f"(x)` does not exist. Hence f is discontinuous at x = 0.