Question

Find the derivative of \[ \frac{d}{dx} \left( \lim_{n \to 1} \frac{x^n - 1}{n+1} \right). \]

• A. 0
• B. \( \frac{1}{2} x \)
• C. \( \frac{1}{2} \)
• D. 1

Hint:

For limits and derivatives combined, simplify the expression first before differentiating.

Answer 1

The Correct Option is B

We need to compute the derivative of the given expression. The limit as \( n \to 1 \) of \( \frac{x^n - 1}{n+1} \) simplifies to \( \frac{x - 1}{2} \). Differentiating this with respect to \( x \), we get: \[ \frac{d}{dx} \left( \frac{x - 1}{2} \right) = \frac{1}{2}. \] Thus, the correct answer is option (B) \( \frac{1}{2} \).