If f " ( x ) is continuous at x = 0 and f " ( 0 ) = 4 , then find the following value of lim x → 0 2 f x - 3 f 2 x + f 4 x x 2

Step-by-step solution

We have, lim x → 0 2 f x - 3 f 2 x + f 4 x x 2 , putting x = 0 , we get 0 0 form. So, Applying L'Hospital rule lim x → 0 2 f x - 3 f 2 x + f 4 x x 2 = lim x → 0 2 f ' x - 6 f ' 2 x + 4 f ' 4 x 2 x again, apply L'Hospital rule, lim x → 0 2 f x - 3 f 2 x + f 4 x x 2 = lim x → 0 2 f ' ' x - 12 f ' ' 2 x + 16 f ' ' 4 x 2 Putting x = 0 , we get lim x → 0 2 f ' ' x - 12 f ' ' 2 x + 16 f ' ' 4 x 2 = 6 f ' ' 0 2 = 12