How to calculate the limit \lim_{x\to1}\lfloor\sin^{-1}(x)\rfloor The function takes the v

Ella Bradshaw

Ella Bradshaw

Answered question

2022-01-27

How to calculate the limit limx1sin1(x)
The function takes the value 1 at x=1 but while approaching 1 from the left side, it takes the value 0 at all points.
The function is not defined at x>1 so the right limit does not exist.
The book mentions 1 as the answer as f(1)=1 but i don't understand why.

Answer & Explanation

goleuedigdp

goleuedigdp

Beginner2022-01-28Added 7 answers

Note that arcsin(1)=π2, not arcsin(1)=1; this means in particular that for values x to the immediate left of 1, also arcsin(x)>1. Thus the function floor(arcsin(x)) is continuous at 1.
However, the answer in the book is bad if it doesn't mention the fact that the function is continuous near x-value 1 and just gives the answer.
helsinka04

helsinka04

Beginner2022-01-29Added 11 answers

You are correct, the limit is only defined as:
limx1arcsin(x)
and since for x1 we have arcsin(x)π2>1 we have that the limit is 1.

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