Q87 / 150 · MHT CET 2024 (03 May Shift 2) · Mathematics
For each $x \in \mathbb{R}$, Let $[x]$ represent greatest integer function, then $\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}$ is equal to
- A0
- B1
- C$\sin 1$
- Correct answerD$-\sin 1$
Solution
$\begin{aligned} & \lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{x} \\ & \text { For } x \rightarrow 0^{-},[x]=-1,|x|=-x \\ \therefore \quad & \lim _{x \rightarrow 0^{-}} \frac{x(-1-x) \sin (-1)}{-x} \\ & =\lim _{x \rightarrow 0^{-}} \frac{-(1+x) \sin (1)}{1} \quad \ldots[x \rightarrow 0, x \neq 0] \\ \therefore & =-(1+0) \sin 1 \\ & =-\sin 1\end{aligned}$
Hence, the correct answer is option: 'D'.