Evaluate : ` underset( x to 0 ) lim (x) ^((1)/(log x))`

Options

A. ` 1`
B. ` 0`
C. ` infty `
D. ` e`

Answer

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

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

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides, we have:
\[ \log y = \lim_{x \to 0} \log\left(x^{\frac{1}{\log x}}\right) \]

### Step 3: Use logarithmic properties
Using the property of logarithms that states \( \log(a^b) = b \cdot \log a \), we can rewrite the limit:
\[ \log y = \lim_{x \to 0} \frac{1}{\log x} \cdot \log x \]

### Step 4: Simplify the expression
Notice that \( \frac{\log x}{\log x} = 1 \) (as long as \( \log x \neq 0 \)), so:
\[ \log y = \lim_{x \to 0} 1 = 1 \]

### Step 5: Exponentiate to solve for \( y \)
Now, we exponentiate both sides to solve for \( y \):
\[ y = e^1 = e \]

### Conclusion
Thus, the value of the limit is:
\[ \lim_{x \to 0} x^{\frac{1}{\log x}} = e \]