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: Understanding the Concept
This problem involves evaluating a limit that includes the greatest integer function. A key property of the greatest integer function is that for any real number \(y\), we have \(y-1<[y] \le y\). We can use this inequality to bound the expression and then apply the Squeeze Theorem (or Sandwich Theorem).
Step 2: Key Formula or Approach
The Squeeze Theorem. We will use the property \(y-1<[y] \le y\).
Applying this to our terms:
- \(\frac{1}{x} - 1<\left[\frac{1}{x}\right] \le \frac{1}{x}\)
- \(\frac{2}{x} - 1<\left[\frac{2}{x}\right] \le \frac{2}{x}\)
We will sum these inequalities, multiply by \(x\), and then take the limit as \(x \to 0^+\).
Step 3: Detailed Explanation
1. Set up the inequalities.
Using the property \(y-1<[y] \le y\), we have:
\[ \left(\frac{1}{x} - 1\right) + \left(\frac{2}{x} - 1\right)<\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] \le \frac{1}{x} + \frac{2}{x} \] \[ \frac{3}{x} - 2<\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] \le \frac{3}{x} \] 2. Multiply by x.
Since we are evaluating the limit as \(x \to 0^+\), \(x\) is a positive number. Therefore, multiplying by \(x\) does not change the direction of the inequalities.
\[ x\left(\frac{3}{x} - 2\right)<x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right]\right) \le x\left(\frac{3}{x}\right) \] \[ 3 - 2x<x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right]\right) \le 3 \] 3. Apply the Squeeze Theorem.
Now we take the limit of all parts of the inequality as \(x \to 0^+\).
- Limit of the lower bound: \[ \lim_{x \to 0^+} (3 - 2x) = 3 - 2(0) = 3 \] - Limit of the upper bound: \[ \lim_{x \to 0^+} (3) = 3 \] Since the expression is squeezed between two functions that both approach 3, by the Squeeze Theorem, the limit of the expression must also be 3.
\[ \lim_{x \to 0^+} x \left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] \right) = 3 \] Step 4: Final Answer
The value of the limit is 3.