if `lim_(x->0)(1+a x)^(b / x)=e^2,` 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. `a=1,b=2`

Explanation

To solve the limit problem given by \( \lim_{x \to 0} (1 + ax)^{\frac{b}{x}} = e^2 \), where \( a \) and \( b \) are natural numbers, we can follow these steps:

### Step 1: Rewrite the limit expression
We start with the expression:
\[ \lim_{x \to 0} (1 + ax)^{\frac{b}{x}} \]
This can be recognized as a limit that resembles the exponential function.

### Step 2: Use the logarithmic limit property
We can take the natural logarithm of the limit:
\[ \ln L = \lim_{x \to 0} \frac{b}{x} \ln(1 + ax) \]
where \( L = \lim_{x \to 0} (1 + ax)^{\frac{b}{x}} \).

### Step 3: Apply the Taylor expansion for \( \ln(1 + u) \)
Using the Taylor series expansion for \( \ln(1 + u) \) around \( u = 0 \):
\[ \ln(1 + ax) \approx ax \text{ as } x \to 0 \]
Thus, we can substitute this into our limit:
\[ \ln L = \lim_{x \to 0} \frac{b}{x} (ax) = \lim_{x \to 0} ab = ab \]

### Step 4: Set the limit equal to the given value
From the problem, we know that:
\[ \ln L = 2 \implies ab = 2 \]

### Step 5: Determine the values of \( a \) and \( b \)
Since \( a \) and \( b \) are natural numbers, we can find the pairs \((a, b)\) that satisfy \( ab = 2\):
- \( (1, 2) \)
- \( (2, 1) \)

### Conclusion
The pairs of natural numbers \( (a, b) \) that satisfy the equation are \( (1, 2) \) and \( (2, 1) \).