$lim x \to 0 \frac{x [x]}{sin | x |}$ , where [.] denotes the greatest integer function, is

0

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1

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not existent

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none of these

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Solution

The correct option is B not existent
$lim x \to 0 \frac{x {x}}{sin | x |}$
For $0 < x \leq 1; [x] = 0$ and for $- 1 < x \leq 0; [x] = - 1$
Now $L . H . L . = lim x \to 0^{-} \frac{x [x]}{sin | x |}$ $= lim x \to 0^{-} \frac{x}{sin | x |} = - 1$

And $R . H . L . = lim x \to 0^{+} \frac{x [x]}{sin | x |}$ $= lim x \to 0^{+} \frac{x (0)}{sin | x |} = 0$
As $L . H . L . \neq R . H . L .$
$∴$ Limit does not exists

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