Let f ( x ) = x 6 + 2 x 4 + x 3 + 2 x + 3 , x ∈ R . Then the natural number n for which lim x → 1 x n f ( 1 ) - f ( x ) x - 1 = 44 is _____ .

Step-by-step solution

Given that: f ( x ) = x 6 + 2 x 4 + x 3 + 2 x + 3 ,   x ∈ R So, f ( 1 ) = 9 lim x → 1 9 x n - x 6 - 2 x 4 - x 3 - 2 x - 3 x - 1 = 44 ⇒ lim x → 1 9 x n - 9 - x 6 + 1 - 2 x 4 + 2 - x 3 + 1 - 2 x + 2 x - 1 = 44 ⇒ lim x → 1 9 x n - 1 x - 1 - x 6 - 1 x - 1 - 2 x 4 - 1 x - 1 - x 3 - 1 x - 1 - 2 x - 1 x - 1 = 44 As we know that, lim x → a x n - a n x - a = n a n - 1 ⇒ 9 n - 6 - 8 - 3 - 2 = 44 ⇒ n = 7