If `lim_(nrarroo) (n.2^(n))/(n(3x-4)^(n)+n.2^(n+1)+2^(n))=1/2` where `"n" epsilonN` then the number of integers in the range of `x` is (a) 0 (b) 2 (c) 3 (d) 1

Options

A. `0`
B. `2`
C. `3`
D. `1`

Explanation

To solve the limit problem, we start with the given expression:

\[ \lim_{n \to \infty} \frac{n \cdot 2^n}{n(3x - 4)^n + n \cdot 2^{n+1} + 2^n} = \frac{1}{2} \]

### Step 1: Simplify the Expression
We can simplify the limit by dividing the numerator and the denominator by \(n \cdot 2^n\):

\[ \lim_{n \to \infty} \frac{1}{\frac{n(3x - 4)^n}{n \cdot 2^n} + \frac{n \cdot 2^{n+1}}{n \cdot 2^n} + \frac{2^n}{n \cdot 2^n}} \]

This simplifies to:

\[ \lim_{n \to \infty} \frac{1}{\frac{(3x - 4)^n}{2^n} + 2 + \frac{1}{n}} \]

### Step 2: Analyze the Limit
As \(n\) approaches infinity, the term \(\frac{1}{n}\) approaches 0. Thus, we have:

\[ \lim_{n \to \infty} \frac{1}{\frac{(3x - 4)^n}{2^n} + 2} \]

### Step 3: Determine the Dominance of Terms
For the limit to equal \(\frac{1}{2}\), we need:

\[ \frac{(3x - 4)^n}{2^n} \to 0 \]

This occurs when \(|3x - 4| < 2\).

### Step 4: Set Up the Inequality
We set up the inequality:

\[ -2 < 3x - 4 < 2 \]

### Step 5: Solve the Inequality
1. Adding 4 to all parts:

\[ 2 < 3x < 6 \]

2. Dividing by 3:

\[ \frac{2}{3} < x < 2 \]

### Step 6: Identify Integer Solutions
The range \(\frac{2}{3} < x < 2\) translates to:

- The lower bound is approximately \(0.67\).
- The upper bound is \(2\).

The only integer value that lies within this range is \(1\).

### Conclusion
Thus, the number of integers in the range of \(x\) is:

\[ \boxed{1} \]