[x] denotes the greatest integer less than or | vedclass

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(A) Given the limit $\lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x+[x])}{2-\{x\}}=	heta$. For $x ightarrow 0^{-}$,we have $[x] = -1$ and $\{x\} =

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(A) Given the limit $\lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x+[x])}{2-\{x\}}=	heta$. For $x ightarrow 0^{-}$,we have $[x] = -1$ and $\{x\} = x - [x] = x - (-1) = x+1$. Substituting these values into the expression: $	heta = \lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x-1)}{2-(x+1)} = \lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x-1)}{1-x}$. As $x ightarrow 0^{-}$,the expression approaches $\frac{\sin ^{-1}(-1)}{1-0} = \frac{-\pi/2}{1} = -\pi/2$. Thus,$	heta = -\pi/2$. Now,we calculate $\sin 	heta + \cos 	heta = \sin(-\pi/2) + \cos(-\pi/2) = -1 + 0 = -1$.

$[x]$ denotes the greatest integer less than or equal to $x$. If $\{x\}=x-[x]$ and $\lim _{x \rightarrow 0^{-}} \frac{\sin ^{-1}(x+[x])}{2-\{x\}}=\theta$,then $\sin \theta+\cos \theta=$

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