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Question: f f(x) = sin [x]/[x], [x] ¹ 0 = 0, [x] = 0 where [x] denotes the greatest integer less than or equal...

f f(x) = sin [x]/[x], [x] ¹ 0 = 0, [x] = 0 where [x] denotes the greatest integer less than or equal to x; thenlim⁡x→0\lim_{x \rightarrow 0}f(x) equals-

A

1

B

0

C

–1

D

None of these

Answer

None of these

Explanation

Solution

[sin⁡[x][x],x∉[0,1)0,x∈[0,1) \left\lbrack \begin{matrix} \frac{\sin\lbrack x\rbrack}{\lbrack x\rbrack}, & x \notin \lbrack 0,1) \\ 0, & x \in \lbrack 0,1) \end{matrix} \right.\

RHL lim⁡x→0+\lim_{x \rightarrow 0^{+}}0 = 0

LHL lim⁡x→0−\lim _ { x \rightarrow 0 ^ { - } } sin⁡[x][x]\frac{\sin\lbrack x\rbrack}{\lbrack x\rbrack}

lim⁡h→0\lim _ { h \rightarrow 0 } sin⁡[−h][−h]\frac{\sin\lbrack - h\rbrack}{\lbrack - h\rbrack}= sin⁡[−1][−1]\frac{\sin\lbrack - 1\rbrack}{\lbrack - 1\rbrack}= sin 1

Limit does not exist.