Question

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

• A. 1
• B. -1
• C. -3
• D. 3
• E. 0

Hint:

For \(x \to 0^+\), \([x] = 0\); for \(x \to 0^-\), \([x] = -1\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For $x \to 0^+$, $x>0$ and very small. Then $[x] = 0$ and $\frac{x}{|x|} = 1$.

Step 2: Detailed Explanation:
As \(x \to 0^+\), \([x] = 0\) and \(\frac{x}{|x|} = \frac{x}{x} = 1\).
\[ \lim_{x \to 0^+} \left[ 2[x] - \frac{x}{|x|} \right] = 2(0) - 1 = -1 \]

Step 3: Final Answer:
The limit is \(-1\).