Question

For each $x \varepsilon R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\displaystyle\lim_{x\to0^{-}} \frac{x\left(\left[x\right] +\left[x\right]\right) \sin\left[x\right]}{\left|x\right|} $ is equal to

• A. $ - \sin 1$
• B. $0$
• C. $1$
• D. $ \sin 1$

Answer 1

Answer: (A)

$\lim_{x\to0^{-} } \frac{x\left(\left[x\right]+\left|x\right|\right)\sin\left[x\right]}{\left|x\right|}$
$ x \to0^{-} $
$ \left[x\right] = - 1 \Rightarrow \lim_{x\to0^{-}} \frac{x\left(-x-1\right)\sin\left(-1\right)}{-x} = -\sin 1 $
$ \left|x \right| = - x$