One-sided limits at 0 of inverse trig functions of the fractional part
This type of question, asked 3 times: 2021 Paper 6, 2021 Paper 7, 1 Feb 2024. All 3 questions below, oldest first.
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Question 1
2021 Paper 6
63
Let α∈R be such that the function f(x)=⎩⎨⎧{x}−{x}3cos−1(1−{x}2)sin−1(1−{x}),α,x=0x=0 is continuous at x=0, where {x}=x−[x] and [x] is the greatest integer less than or equal to x. Then:
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Question 2
2021 Paper 7
80
The value of limx→0+x−x3cos−1(x−[x]2)⋅sin−1(x−[x]2), where [x] denotes the greatest integer ≤x is:
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Question 3
1 Feb 2024
85
Let {x} denote the fractional part of x and f(x)={x}−{x}3cos−1(1−{x}2)sin−1(1−{x}),x=0. If L and R respectively denotes the left hand limit and the right hand limit of f(x) at x=0, then π232(L2+R2) is equal to
