Question

Let \([a]\) denote the greatest integer less than or equal to \(a\). Then \[ \lim_{x\to 0^{+}} x\left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] \right) \] is equal to ________.

• A. 2
• B. 1
• C. 3
• D. 0
• E. 4

Hint:

$\lim_{x\to 0} x[\frac{k}{x}] = k$.

Answer 1

The Correct Option is C

Step 1: Concept
Use the property that $a - 1 < [a] \le a$.

Step 2: Meaning
The expression is $x[\frac{1}{x}] + x[\frac{2}{x}]$.

Step 3: Analysis
As $x \to 0^{+}$, $x[\frac{1}{x}] \to 1$ and $x[\frac{2}{x}] \to 2$ using the Squeeze Theorem.

Step 4: Conclusion
The sum is $1 + 2 = 3$. Final Answer: (C)