The equivalent definition of the function `f(x)=lim_(n to oo)(x^(n)-x^(-n))/(x^(n)+x^(-n)), x gt 0` , is

Options

A. `f(x)={{:(-1,0 lt x le1),(1, x gt 1):}`
B. `f(x)={{:(-1,0 lt x lt 1),(1, x ge 1):}`
C. `f(x)={{:(-1,0 lt x lt 1),(0, x=1),(1,x gt 1):}`
D. none of these

Answer

To find the equivalent definition of the function \( f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}} \) for \( x > 0 \), we can analyze the limit based on the value of \( x \).

### Step-by-Step Solution:

1. **Rewrite the Function**:
\[ f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}} \]

2. **Consider Different Cases for \( x \)**:
We will analyze the function for three cases: \( 0 < x < 1 \), \( x = 1 \), and \( x > 1 \).

3. **Case 1: \( 0 < x < 1 \)**:
- As \( n \to \infty \), \( x^n \to 0 \) and \( x^{-n} \to \infty \).
- Thus, we have:
\[ f(x) = \lim_{n \to \infty} \frac{0 - x^{-n}}{0 + x^{-n}} = \lim_{n \to \infty} \frac{-x^{-n}}{x^{-n}} = -1 \]

4. **Case 2: \( x = 1 \)**:
- Here, both \( x^n \) and \( x^{-n} \) equal 1.
- Thus, we have:
\[ f(1) = \lim_{n \to \infty} \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0 \]

5. **Case 3: \( x > 1 \)**:
- As \( n \to \infty \), \( x^n \to \infty \) and \( x^{-n} \to 0 \).
- Thus, we have:
\[ f(x) = \lim_{n \to \infty} \frac{x^n - 0}{x^n + 0} = \lim_{n \to \infty} \frac{x^n}{x^n} = 1 \]

6. **Conclusion**:
- We can summarize the results:
- If \( 0 < x < 1 \), then \( f(x) = -1 \).
- If \( x = 1 \), then \( f(x) = 0 \).
- If \( x > 1 \), then \( f(x) = 1 \).
- Therefore, the equivalent definition of the function can be expressed as:
\[ f(x) = \begin{cases} -1 & \text{if } 0 < x < 1 \\ 0 & \text{if } x = 1 \\ 1 & \text{if } x > 1 \end{cases} \]