Question

STATEMENT-1 : $\underset{x\to 0}{lim}{\sin }^{-1}\left\{x\right\}$ does not exist (where {.} denotes fractional part function).
STATEMENT-2 : {x} is discontinuous at $x=0$.

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• C. STATEMENT-1 is True, STATEMENT-2 is False
• D. STATEMENT-1 is False, STATEMENT-2 is True

Answer 1

The correct option is B STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 : $\underset{x\to 0}{lim}{\sin }^{-1}\left\{x\right\}$
LHL $=\underset{x\to 0^{-}}{lim}{\sin }^{-1}\left\{x\right\}={\sin }^{-1}\left(1\right)=\frac{\pi }{2}$
RHL $=\underset{x\to 0^{+}}{lim}{\sin }^{-1}\left\{x\right\}={\sin }^{-1}\left(0\right)=0$
Clearly LHL$\ne$RHL, Thus given limit does not exist.
Also we know that $\left\{x\right\}$ is discontinuous at all integral points
Therefore both statements are correct but 2 is not explanation of 1.