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Solve \lim _{x \rightarrow 0}\left[\frac{\sin |x|}{|x|}\right] \text {, where }[.] denotes greatest integer function. 

Option: 1

0


Option: 2

1


Option: 3

2

 


Option: 4

none of these 


Answers (1)

best_answer

We have , 

\lim _{x \rightarrow 0}\left[\frac{\sin |x|}{|x|}\right]

We know, \frac{\sin x}{x} \rightarrow 1 \text { as } x \rightarrow 0

Or         \frac{\sin |x|}{|x|} \rightarrow 1 \text { as } x \rightarrow 0 \text { from right or left }

I.e., 

\text { at } x=0+h \text { or } x=0-h \text { and less than one, }

\frac{\sin |x|}{|x|}=0 \text { as } \frac{\sin |x|}{|x|}<1

\Rightarrow from equation (i) whether we find R H L or L H L 

\begin{array}{ll} & \lim _{x \rightarrow 0}\left[\frac{\sin |x|}{|x|}\right]=\lim _{x \rightarrow 0} 0 \Rightarrow 0 . \\ \\\therefore & \quad \lim _{x \rightarrow 0}\left[\frac{\sin |x|}{|x|}\right] \text {, exists and is } 0 . \end{array}

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Nehul

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