Let f x = g x e 1 x   -   e - 1 x e 1 x   +   e - 1 x , where g is a continuous function then lim x → 0 ⁡ f x exists if

Step-by-step solution

lim x → 0 + e 1 x - e - 1 x e 1 x + e - 1 x =   lim x → 0 + 1 - e - 2 x 1 + e - 2 x = 1 lim x → 0 - e 1 x - e - 1 x e 1 x + e - 1 x =   lim x → 0 - e 2 x - 1 e 2 x + 1 = - 1 Hence lim x → 0 ⁡ f x exists if g x = x h x where h ( x ) is a continuous function. If g x = a   a ≠ 0 then lim x → 0 + f x = a and lim x → 0 - f x = - a . Thus lim x → 0 ⁡ f x does not exist if g x = x + 2 or x 4 + 4 or is a constant function.