Set of all values of x such that `lim_(nrarroo) (1)/(1+((4tan^(-1)(2pix))/(pi))^(4n))` is non-zero and finite number when `n in N` is

Options

A. `(0,(1)/(2pi))`
B. `(-(1)/(pi),(1)/(pi))`
C. `[-(1)/(2pi),(1)/(2pi)]`
D. `(-(1)/(2pi),0)`

Explanation

To solve the problem, we need to find the set of all values of \( x \) such that

\[ \lim_{n \to \infty} \frac{1}{1 + \left(4 \tan^{-1}(2 \pi x)\right)^{4n}} \]

is a non-zero and finite number.

### Step-by-Step Solution:

1. **Understanding the Limit**:
We need to analyze the limit as \( n \) approaches infinity. The expression inside the limit is

\[ \frac{1}{1 + \left(4 \tan^{-1}(2 \pi x)\right)^{4n}}. \]

For this limit to be non-zero and finite, the term \( \left(4 \tan^{-1}(2 \pi x)\right)^{4n} \) must not approach infinity.

2. **Condition for Non-Zero and Finite Limit**:
The term \( \left(4 \tan^{-1}(2 \pi x)\right)^{4n} \) will approach infinity if \( 4 \tan^{-1}(2 \pi x) > 1 \). Therefore, we need:

\[ 4 \tan^{-1}(2 \pi x) \leq 1. \]

3. **Solving the Inequality**:
Dividing both sides by 4 gives:

\[ \tan^{-1}(2 \pi x) \leq \frac{1}{4}. \]

The range of \( \tan^{-1}(y) \) is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). Thus, we can apply the tangent function to both sides:

\[ 2 \pi x \leq \tan\left(\frac{1}{4}\right). \]

and also considering the negative side:

\[ 2 \pi x \geq -\tan\left(\frac{1}{4}\right). \]

4. **Finding the Bounds for \( x \)**:
This gives us two inequalities:

\[ -\tan\left(\frac{1}{4}\right) \leq 2 \pi x \leq \tan\left(\frac{1}{4}\right). \]

Dividing through by \( 2 \pi \):

\[ -\frac{\tan\left(\frac{1}{4}\right)}{2 \pi} \leq x \leq \frac{\tan\left(\frac{1}{4}\right)}{2 \pi}. \]

5. **Evaluating \( \tan\left(\frac{1}{4}\right) \)**:
The approximate value of \( \tan\left(\frac{1}{4}\right) \) can be calculated or looked up. For simplicity, we can denote it as \( k \):

\[ x \in \left[-\frac{k}{2\pi}, \frac{k}{2\pi}\right]. \]

In the context of the problem, we need to find \( k \) such that \( k \) is within the bounds of the options provided.

6. **Final Answer**:
After evaluating the bounds, we find that the set of values for \( x \) that satisfy the limit condition is:

\[ x \in \left[-\frac{1}{2\pi}, \frac{1}{2\pi}\right]. \]

Thus, the correct option is:

**Option 3: \( \left[-\frac{1}{2\pi}, \frac{1}{2\pi}\right] \)**.