If `lim_(x to 0) (1+ax)^(b//x)=e^4` , where a and b are natural numbers then

Options

A. a=4,b=2
B. a=8,b=4
C. a=16,b=8
D. none of these

Answer

To solve the limit problem given by \( \lim_{x \to 0} (1 + ax)^{\frac{b}{x}} = e^4 \), where \( a \) and \( b \) are natural numbers, we can follow these steps:

### Step 1: Recognize the Limit Form
We know that as \( x \to 0 \), the expression \( (1 + ax)^{\frac{b}{x}} \) approaches the form \( (1 + 0)^{\infty} \), which is an indeterminate form. We can use the exponential limit property.

### Step 2: Rewrite the Limit
Using the property of limits, we can rewrite the limit as:
\[ \lim_{x \to 0} (1 + ax)^{\frac{b}{x}} = e^{\lim_{x \to 0} \frac{b}{x} \ln(1 + ax)} \]

### Step 3: Simplify the Logarithm
Next, we need to simplify \( \ln(1 + ax) \) as \( x \to 0 \). Using the Taylor expansion, we have:
\[ \ln(1 + ax) \approx ax \text{ as } x \to 0 \]

### Step 4: Substitute Back into the Limit
Substituting this approximation into our limit gives:
\[ \lim_{x \to 0} \frac{b}{x} \ln(1 + ax) \approx \lim_{x \to 0} \frac{b}{x} (ax) = \lim_{x \to 0} ab = ab \]

### Step 5: Set the Limit Equal to \( e^4 \)
From the original problem, we have:
\[ e^{ab} = e^4 \]
This implies:
\[ ab = 4 \]

### Step 6: Find Natural Number Solutions
Now, we need to find pairs of natural numbers \( (a, b) \) such that \( ab = 4 \). The pairs are:
- \( (1, 4) \)
- \( (2, 2) \)
- \( (4, 1) \)

### Conclusion
Thus, the possible values of \( (a, b) \) are \( (1, 4), (2, 2), (4, 1) \).