<p>lim
x→0+
cos−1(x −[x]2) · sin−1(x −[x]2)
x −x3
,
where [x] denotes the greatest integer less than or equal to x, is:</p>
Step-by-Step Solution
Key Concept: For x \to 0+, the GIF part becomes constant immediately.
<p>As x \to 0+, we have [x] = 0. So the limit becomes</p> lim x\to 0+ cos-1 x \cdot sin-1 x x -x3 . Now, near x = 0, cos-1 x = \pi 2 -sin-1 x. Hence, cos-1 x \cdot sin-1 x = \pi 2 -sin-1 x sin-1 x. Since sin-1 x ∼x as x \to 0, cos-1 x \cdot sin-1 x ∼\pi 2 x. Also, x -x3 = x(1 -x2) ∼x. Therefore, lim x\to 0+ cos-1 x \cdot sin-1 x x -x3 = \pi 2 . Shortcut / Fast View On the right of 0, replace [x] by 0 first. Then use cos-1 x = \pi 2 -sin-1 x.
Correct Answer: (4)