Question

Let \(x \to 1\) \(\frac{x^x - x}{x - 1 - \log x}\) = _______

• A. 0
• B. 1
• C. 2
• D. 1/2

Hint:

When solving limits involving indeterminate forms like \( \frac{0}{0} \), apply L'Hopital's Rule to simplify the expression.

Answer 1

The Correct Option is C

To solve this, we can apply L'Hopital's Rule. We need to differentiate the numerator and denominator separately:
- The numerator is \( x^x - x \), which requires implicit differentiation.
- The denominator is \( x - 1 - \log x \), which can be simplified by applying basic differentiation rules.
Applying L'Hopital's Rule by differentiating the numerator and denominator with respect to \( x \), we get the following: \[ \lim_{x \to 1} \frac{x^x - x}{x - 1 - \log x} = 2 \] Thus, the correct answer is \( 2 \).