If lim x → 0 ⁡ sin - 1 ⁡ x - tan - 1 ⁡ x x 3 equals L , then the value of ( 4 L + 1 ) is

Step-by-step solution

Let L = lim x → 0 ⁡ sin - 1 ⁡ x - tan - 1 ⁡ x x 3 0 0 f o r m = lim x → 0 ⁡ 1 1 - x 2 - 1 1 + x 2 3 x 2 L ′ H o s p i t a l r u l e = 1 3 lim x → 0 ⁡ 1 x 2 1 + x 2 - 1 - x 2 1 + x 2 1 - x 2 = 1 3 lim x → 0 ⁡ 1 x 2 1 + x 2 2 - 1 - x 2 1 + x 2 1 - x 2 . 1 1 + x 2 + 1 - x 2 = 1 3 lim x → 0 ( 3 + x 2 ) ( 1 + x 2 ) 1 − x 2 1 [ ( 1 + x 2 ) + 1 − x 2 ] = 1 3 3 2 = 1 2