Let f : R → R be defined as f x = x 3 ( 1 - cos 2 x ) 2 log e 1 + 2 x e - 2 x 1 - x e - x 2 , x ≠ 0 α , x = 0 If f is continuous at x = 0 , then α is equal to:

Step-by-step solution

A function f x is continuous at a point x = a , if lim x → a f x = f a . Thus, for continuity of the given function, we have lim x → 0 f x = α ⇒   lim x → 0 x 3 ( 1 - cos 2 x ) 2 log e 1 + 2 x e - 2 x 1 - x e - x 2 = α Using, cos 2 x = 1 - 2 sin 2 x ,   log e m n = log e m - log e n &   log e m n = n log e m , we get lim x → 0 x 3 2 sin 2 x 2 log e 1 + 2 x e - 2 x - log e 1 - x e - x 2 = α ⇒   lim x → 0 x 3 4 sin 4 x log e 1 + 2 x e -