If `lim_(xrarr0) [1+x+(f(x))/(x)]^(1//x)=e^(3)", then "lim_(xrarr0) [1+(f(x))/(x)]^(1//x)=`

Options

A. e
B. `e^(2)`
C. `e^(3)`
D. none of these

Explanation

To solve the problem step by step, we start with the given limit:

**Given:**
\[ \lim_{x \to 0} \left[1 + x + \frac{f(x)}{x}\right]^{\frac{1}{x}} = e^3 \]

We need to find:
\[ \lim_{x \to 0} \left[1 + \frac{f(x)}{x}\right]^{\frac{1}{x}} \]

### Step 1: Analyze the Given Limit
We know that the limit can be expressed in the form \( e^{\lim_{x \to 0} g(x)} \) where \( g(x) \) is some function we need to determine.

From the given limit, we can rewrite it as:
\[ \lim_{x \to 0} \left[1 + x + \frac{f(x)}{x}\right]^{\frac{1}{x}} = e^{\lim_{x \to 0} \left( \left(1 + x + \frac{f(x)}{x}\right) - 1 \right) \cdot \frac{1}{x}} \]

### Step 2: Simplify the Expression Inside the Limit
Now, we simplify:
\[ 1 + x + \frac{f(x)}{x} - 1 = x + \frac{f(x)}{x} \]
Thus, we have:
\[ \lim_{x \to 0} \left( x + \frac{f(x)}{x} \right) \cdot \frac{1}{x} = \lim_{x \to 0} \left( 1 + \frac{f(x)}{x^2} \right) \]

### Step 3: Set Up the Limit Equation
From the original limit, we know:
\[ \lim_{x \to 0} \left( 1 + \frac{f(x)}{x^2} \right) = 3 \]
This implies:
\[ \lim_{x \to 0} \frac{f(x)}{x^2} = 2 \]

### Step 4: Find the Desired Limit
Now, we need to find:
\[ \lim_{x \to 0} \left[1 + \frac{f(x)}{x}\right]^{\frac{1}{x}} \]
We can express this limit similarly:
\[ \lim_{x \to 0} \left[1 + \frac{f(x)}{x}\right]^{\frac{1}{x}} = e^{\lim_{x \to 0} \left( \frac{f(x)}{x} - 1 \right) \cdot \frac{1}{x}} \]

### Step 5: Substitute the Known Limit
We know from earlier that:
\[ \lim_{x \to 0} \frac{f(x)}{x^2} = 2 \implies \lim_{x \to 0} f(x) = 2x^2 \]
Thus:
\[ \lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to 0} \frac{2x^2}{x} = \lim_{x \to 0} 2x = 0 \]

### Step 6: Final Calculation
Now we can substitute this back into our limit:
\[ \lim_{x \to 0} \left[1 + \frac{f(x)}{x}\right]^{\frac{1}{x}} = e^{\lim_{x \to 0} \left( 0 - 1 \right) \cdot \frac{1}{x}} = e^{-\infty} = 0 \]

### Conclusion
Thus, the final answer is:
\[ \lim_{x \to 0} \left[1 + \frac{f(x)}{x}\right]^{\frac{1}{x}} = 0 \]