Question

Evaluate the limit: \[ \lim_{x \to a} \frac{\log{(x^{a-1})}}{x-a} \]

• A. 0
• B. \( \infty \)
• C. \( \log_a e \)
• D. \( \frac{1}{a} \log_a e \)

Hint:

For limits involving logarithms and powers, use L'Hopital's Rule when the limit results in an indeterminate form like \( \frac{0}{0} \).

Answer 1

The Correct Option is D

The limit can be simplified using L'Hopital's Rule: \[ \lim_{x \to a} \frac{\log{(x^{a-1})}}{x-a} = \frac{1}{a} \log_a e \]