Let for all `x gt 0, f(x)=lim_(nrarroo)n^((x^((1)/(n))-1))`, then

Options

A. `f(x)+f((1)/(x))=1`
B. `f(xy)=f(x)+f(y)`
C. `f(xy)=xf(y)+yf(x)`
D. `f(xy)=xf(x)+yf(x)`

Answer

To solve the problem, we need to evaluate the limit:

\[ f(x) = \lim_{n \to \infty} n \left( x^{\frac{1}{n}} - 1 \right) \]

### Step 1: Rewrite the limit
We can rewrite \( n \) as \( \frac{1}{t} \) where \( t = \frac{1}{n} \). As \( n \to \infty \), \( t \to 0 \). Thus, we can rewrite the limit as:

\[ f(x) = \lim_{t \to 0} \frac{x^{t} - 1}{t} \]

### Step 2: Recognize the form of the limit
The expression \( \frac{x^{t} - 1}{t} \) as \( t \to 0 \) is in the form of the derivative of \( x^t \) at \( t = 0 \). We can apply L'Hôpital's Rule since it is an indeterminate form \( \frac{0}{0} \).

### Step 3: Apply L'Hôpital's Rule
Differentiating the numerator and denominator with respect to \( t \):

- The derivative of the numerator \( x^{t} - 1 \) is \( x^{t} \ln(x) \).
- The derivative of the denominator \( t \) is \( 1 \).

Thus, we have:

\[ f(x) = \lim_{t \to 0} \frac{x^{t} \ln(x)}{1} = \lim_{t \to 0} x^{t} \ln(x) \]

### Step 4: Evaluate the limit
As \( t \to 0 \), \( x^{t} \) approaches \( 1 \):

\[ f(x) = 1 \cdot \ln(x) = \ln(x) \]

### Conclusion
Therefore, we find that:

\[ f(x) = \ln(x) \]

### Step 5: Verify the property
Now, we need to check if this function satisfies the property \( f(xy) = f(x) + f(y) \):

\[ f(xy) = \ln(xy) = \ln(x) + \ln(y) = f(x) + f(y) \]

This confirms that the function satisfies the logarithmic property.

### Final Answer
The function \( f(x) \) is such that:

\[ f(xy) = f(x) + f(y) \]