Let f x = x − 1 , x is even , 2 x , x is odd , x ∈ N . If for some a ∈ N , f f f a = 21 , then lim x → a − x 3 a − x a , where t denotes the greatest integer less than or equal to t , is equal to:

Step-by-step solution

Given: f x = x − 1 ; x is even 2 x : x is odd And f f f a = 21 Case-I: If a is even ⇒ f a = a − 1 = Odd ⇒ f f a = 2 a − 1 = Even ⇒ f f f a = 2 a − 3 = 21 ⇒ a = 12 Case-II: If a is odd ⇒ f a = 2 a = Even ⇒ f f a = 2 a − 1 = Odd ⇒ f f f a = 4 a − 2 = 21 (Not possible) Hence, a = 12 Now, lim x → 12 − x 3 12 − x 12 = lim x → 12 − x 3 12 − lim x → 12 − x 12 ⇒ lim x → 12 − x 3 12 − x 12 = 144 - 0 ⇒ lim x → 12 − x 3 12 − x 12 = 144