The function f : R → R defined by f x = lim n → ∞ cos 2 π x - x 2 n sin x - 1 1 + x 2 n + 1 - x 2 n is continuous for all x in

Step-by-step solution

Given f x = lim n → ∞ cos 2 π x - x 2 n sin x - 1 1 + x 2 n + 1 - x 2 n = lim n → ∞ cos 2 π x - x 2 n sin x - 1 1 + x x 2 n - x 2 n Now for - 1 < x < 1 , as 0 < x 2 < 1 ⇒ lim n → ∞ x 2 n → 0 i.e. f x = cos 2 π x Now again rewriting f x = lim n → ∞ x 2 - n cos 2 π x - sin x - 1 x 2 - n + x - 1 For x > 1   or   x < - 1 , lim n → ∞ x - 2 n → 0 i.e. f x = - sin x - 1 x - 1 For x = ± 1