A) 1
B) 0
C) \[sin\text{ }1\]
D) \[-sin\text{ }1\]
Correct Answer: D
Solution :
\[\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\,\frac{x([x]+\left| x \right|)\,\,sin\,[x]}{\left| x \right|}\] \[\underset{h\to 0}{\mathop{\lim }}\,\,\frac{(0-h)([0-h]+\left| 0-h \right|)sin\,[0-h]}{\left| 0-h \right|}\] \[=\,\,\,\underset{h\to 0}{\mathop{\lim }}\,\,\frac{-h(-1+h)sin(-1)}{h}\] \[=\,\,\,\underset{h\to 0}{\mathop{\lim }}\,\,(-1+h)\,sin(1)\,\,=\,\,-sin1\]
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