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  • Choose the correct answer out of 4 options given against each Question lim┬(x→0^- ) (sin⁡[x]/[x] ),xnotequalto 0 where [.] denotes the greatest integer function, then lim f(x) =

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

If \begin{aligned} f(x)=\frac{\sin [x]}{[x]}, &[x] \neq 0 \\ 0, &[x]=0 \end{aligned}  where [.] denotes the greatest integer function, then \lim_{x\rightarrow 0}f(x) is equal to


A. 1
B. 0
C. –1
D. None of these

Answers (1)

 LHL=\mathop{\lim }_{x \rightarrow \mathop{0}^{-}} \left( \frac{\sin \left[ x \right] }{ \left[ x \right] } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{\sin \left[ 0-h \right] }{ \left[ 0-h \right] } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{\sin \left[ -h \right] }{ \left[ -h \right] } \right) \\=\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{\sin \left( -1 \right) }{-1} \right) =\sin 1 \\ \\

    RHL=\mathop{\lim }_{x \rightarrow \mathop{0}^{+}} \left( \frac{\sin \left[ x \right] }{ \left[ x \right] } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{+}} \left( \frac{\sin \left[ 0+h \right] }{ \left[ 0+h \right] } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{+}} \left( \frac{\sin \left[ h \right] }{ \left[ h \right] } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{+}} \left( \frac{\sin \left( 0 \right) }{0} \right) \\ \\

Limit doesn’t exist   

Hence, the answer is option D

 

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