Evaluate :
`lim_(xto0) (1+ax)^(1//x)`

Answer

To evaluate the limit \( \lim_{x \to 0} (1 + ax)^{\frac{1}{x}} \), we can follow these steps:

### Step 1: Define the limit
Let \( y = \lim_{x \to 0} (1 + ax)^{\frac{1}{x}} \).

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides, we get:
\[ \log y = \log \left( \lim_{x \to 0} (1 + ax)^{\frac{1}{x}} \right) \]
Using the property of logarithms, we can bring the exponent down:
\[ \log y = \lim_{x \to 0} \frac{1}{x} \log(1 + ax) \]

### Step 3: Apply L'Hôpital's Rule
As \( x \to 0 \), both the numerator \( \log(1 + ax) \) and the denominator \( x \) approach 0. Therefore, we can apply L'Hôpital's Rule:
\[ \log y = \lim_{x \to 0} \frac{\frac{d}{dx} \log(1 + ax)}{\frac{d}{dx} x} \]

### Step 4: Differentiate the numerator and denominator
The derivative of \( \log(1 + ax) \) is:
\[ \frac{d}{dx} \log(1 + ax) = \frac{a}{1 + ax} \]
The derivative of \( x \) is simply \( 1 \). Thus, we have:
\[ \log y = \lim_{x \to 0} \frac{a}{1 + ax} \]

### Step 5: Evaluate the limit
Now we can evaluate the limit:
\[ \log y = \frac{a}{1 + a \cdot 0} = \frac{a}{1} = a \]

### Step 6: Solve for \( y \)
Now, we exponentiate both sides to solve for \( y \):
\[ y = e^{\log y} = e^a \]

### Final Result
Thus, the limit is:
\[ \lim_{x \to 0} (1 + ax)^{\frac{1}{x}} = e^a \]

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