Question

Choose the most appropriate option. 
The value of \(\lim_{x \to a} \frac{\log x - 1}{x - a}\) is equal to

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

Hint:

This type of limit is a definition of the derivative for the natural logarithm function at a point.

Answer 1

The Correct Option is D

We are asked to compute the limit \( \lim_{x \to a} \frac{\log x - 1}{x - a} \).
Let \( f(x) = \log x \).
We recognize that this is a standard limit involving the derivative of \( \log x \) at \( x = a \).
\[ \lim_{x \to a} \frac{\log x - \log a}{x - a} = \frac{1}{a} \] Therefore, the correct answer is \( \frac{1}{a} \log_a e \).