Let α be a positive real number. Let f : ℝ → ℝ and g : α , ∞ → ℝ be the functions defined by f x = sin π x 12 and g x = 2 log e x - α log e e x - e α . Then the value of lim x → α + f g x is ______.

Step-by-step solution

lim x → α + g x = lim x → α + 2 log e x - α log e e x - e α = lim x → α + 2 x - α 1 2 x 1 e x - e α 1 2 x e x = lim x → α + e x - e α x - α 1 e x · 2 = 2 e α lim x → α + e x · 1 2 x 1 2 x = 2 lim x → α + f g x = f lim x → α + g x = f 2 = sin π 6 = 1 2 = 0 . 50