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Left Hand Limit

For each x ∈ℝ...

Question

For each $x \in R$ , let $[x]$ be the greatest integer less than or equal to $x$ . Then $lim x \to 0^{-} \frac{x ([x] + | x |) sin [x]}{| x |}$ is equal to :

0

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sin 1

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1

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- sin 1

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Solution

The correct option is D $- sin 1$
$lim x \to 0^{-} \frac{x ([x] + | x |) sin [x]}{| x |}$
We know,
As $x \to 0^{-}, [x] = - 1$ and $| x | = - x$

So, $lim x \to 0^{-} \frac{x (- 1 - x) sin (- 1)}{- x}$

$= lim x \to 0^{-} - \frac{(1 + x) sin 1}{1}$

$= - \frac{(1 + 0) sin 1}{1}$

$= - sin 1$

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