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  • Choose the correct answer out of 4 options given against each Question 〖lim〗┬(x→0) ((|sin⁡x |)/x) is A. 1 B. –1 C. does not exist D. None of these

Choose the correct answer out of 4 options given against each Question

\mathop{\lim }_{x \rightarrow 0} \left( \frac{ \vert \sin x \vert }{x} \right) ~ is
A. 1
B. –1
C. does not exist
D. None of these

Answers (1)

   \\ LHL=\mathop{\lim }_{x \rightarrow \mathop{0}^{-}} \left( \frac{ \vert \sin x \vert }{x} \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{ \vert \sin \left( 0-h \right) \vert }{ \left( 0-h \right) } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{-\sin \left( -h \right) }{-h} \right) \\=-1 \\ \\ RHL=\mathop{\lim }_{x \rightarrow \mathop{0}^{+}} \left( \frac{ \vert \sin x \vert }{x} \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{+}} \left( \frac{ \vert \sin \left( 0+h \right) \vert }{ \left( 0+h \right) } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{+}} \left( \frac{\sin \left( h \right) }{h} \right) =1 \\ \\
    \text{ So, limit does}n^{'}\text{t exists} \\ \\ \text{Hence, the answer is option C} \\ \\

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