Discussion
If [x] denotes the greatest integer less than or equal to x for \(x \in \mathbb{R}\), then the value of \(\lim_{x \to 0^{-}}\left(2[x] - \frac{x}{|x|}\right)\) is equal to
Answer: B
The limit is \(-1\).
For \(x\) slightly less than 0, say \(-0.1\), the greatest integer \([x]\) is \(-1\), and \(|x| = -x\), so \(\frac{x}{|x|} = -1\):
\[ \begin{aligned} \lim_{x \to 0^{-}}\left(2[x] - \frac{x}{|x|}\right) \\ &= 2(-1) - (-1) \\ &= -1 \end{aligned} \]
Using \([x] = 0\) (the value at 0 itself) gives 1, and taking \(\frac{x}{|x|} = 1\) gives \(-3\); both are the right-hand or wrong-sign readings.
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