Solve the following limit.

Question

$\lim_{x\to 1} [\sin^{-1} x]$ ; where [.] is the 'Greatest Integer Function'.

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The left hand limit will be $ [π/2]$ = $1$. But how can there be a right hand limit (as $ 'x'$ can't take values greater than $1$)? The answer in my textbook is given as $1$. But how can the limit exist when there is no right hand limit because for a limit to exist, LHL should be equal to the RHL.

Answer 1

Note that $\arcsin x$ is only defined for $x\in [-1,1]$ thus the limit is to be assumed as $x\to1^-$.

Answer 2

Yes, you are right to question the book, it is wrong, the limit does not exist, only RHS limit exists, but not the LHS limit ( In complex domain LHS limit also exists, but that will be part of Complex analysis course).

For your purposes the book is wrong, see https://en.wikipedia.org/wiki/One-sided_limit