A) equals 1
B) equals 0
C) does not exist
D) equals? 1
Correct Answer: B
Solution :
Consider \[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{x-\sin x}{x} \right)\sin \left( \frac{1}{x} \right)\] \[=\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{x\left( 1-\frac{\sin x}{x} \right)}{x} \right]\times =\underset{x\to 0}{\mathop{\lim }}\,\sin \left( \frac{1}{x} \right)\] \[=\underset{x\to 0}{\mathop{\lim }}\,\left[ 1-\frac{\sin x}{x} \right]\times \underset{x\to 0}{\mathop{\lim }}\,\sin \left( \frac{1}{x} \right)\] \[=\left[ 1-\underset{x\to }{\mathop{\lim }}\,\frac{\sin x}{x} \right]\times \underset{x\to 0}{\mathop{\lim }}\,\sin \left( \frac{1}{x} \right)\] \[=0\times \underset{x0}{\mathop{\lim }}\,\sin \left( \frac{1}{x} \right)=0\]
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