If lim x → 0 e a x - cos ( b x ) - c x e - c x 2 1 - cos ( 2 x ) = 17 , then 5 a 2 + b 2 is equal to

Step-by-step solution

Given that ⇒ lim x → 0 e a x - cos ( b x ) - c x 2 e - c x 1 - cos 2 x = 17 We know that e a x = 1 + a x + ( a x ) 2 2 ! + … and cos b x = 1 - ( b x ) 2 2 ! + … ⇒ lim x → 0 1 + a x + a x 2 2 ! + … - 1 - b x 2 2 ! + … - c x 2 1 - c x + c x 2 2 ! - … 1 - cos 2 x 2 x 2 × 4 x 2 = 17 ⇒ lim x → 0 a - c 2 x + a 2 + b 2 + c 2 2 x 2 + … 1 2 × 4 x 2 = 17 Now for limit to exist, a - c 2 = 0 ⇒ c = 2 a ⇒ lim x → 0 a 2 + b 2 + c 2