If `f(x)=(1+x)^(5//x)` is continuous at x=0, then what is the value of f(0)?

Options

A. 0
B. 1
C. `oo`
D. `e^(5)`

Explanation

To find the value of \( f(0) \) for the function \( f(x) = (1+x)^{\frac{5}{x}} \) given that it is continuous at \( x = 0 \), we need to evaluate the limit of \( f(x) \) as \( x \) approaches 0.

### Step-by-Step Solution:

1. **Identify the Function**:
The function is given as:
\[ f(x) = (1+x)^{\frac{5}{x}} \]

2. **Check Continuity at \( x = 0 \)**:
Since \( f(x) \) is continuous at \( x = 0 \), we need to find:
\[ f(0) = \lim_{x \to 0} f(x) = \lim_{x \to 0} (1+x)^{\frac{5}{x}} \]

3. **Rewrite the Limit**:
We can rewrite the limit as:
\[ \lim_{x \to 0} (1+x)^{\frac{5}{x}} = \lim_{x \to 0} \left( (1+x)^{\frac{1}{x}} \right)^5 \]

4. **Evaluate the Inner Limit**:
We need to evaluate:
\[ \lim_{x \to 0} (1+x)^{\frac{1}{x}} \]
This limit is a well-known limit that approaches \( e \) as \( x \) approaches 0.

5. **Combine the Results**:
Therefore, we have:
\[ \lim_{x \to 0} (1+x)^{\frac{5}{x}} = \left( \lim_{x \to 0} (1+x)^{\frac{1}{x}} \right)^5 = e^5 \]

6. **Conclusion**:
Thus, since \( f(x) \) is continuous at \( x = 0 \), we find:
\[ f(0) = e^5 \]

### Final Answer:
The value of \( f(0) \) is \( e^5 \).