If lim _(x → 0) (sin ⁻¹x - tan ⁻¹x)/3x³ is equal to L, then the value of (6L + 1) is

Solution

Maths sir · 3 steps

Expand both inverse functions up to x³: the x terms cancel and the x³ terms decide L.

1. Expand each function. Near x = 0, sin⁻¹ x = x + x³/6 + … and tan⁻¹ x = x - x³/3 + …. The denominator is 3x³, so you only need terms up to x³.
2. Subtract. The x terms cancel. What is left is x³/6 + x³/3 = x³/2, plus higher powers of x that vanish in the limit.

   L = lim(x → 0) (x³/2 + …)/3x³ = 1/2 × 1/3 = 1/6

3. So. 6L + 1 = 6 × 1/6 + 1 = 2, which is option (d).

The idea to keep. When the leading terms cancel in a 0/0 limit, expand each function up to the power in the denominator. Know the x³ terms of sin x, tan x, sin⁻¹ x and tan⁻¹ x by heart.

Answer from the NTA final answer key. Solution by PrepWiser.