Question:
Choose the most appropriate option.
The value of \(\lim_{x \to a} \frac{\log x - 1}{x - a}\) is equal to
Choose the most appropriate option.
The value of \(\lim_{x \to a} \frac{\log x - 1}{x - a}\) is equal to
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This type of limit is a definition of the derivative for the natural logarithm function at a point.
Updated On: Dec 11, 2025
- \( \frac{1}{a} \)
- \( a \)
- \( \log_a e \)
- \( \frac{1}{a} \log_a e \)
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The Correct Option is D
Solution and Explanation
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 \( \frac{1}{a} \log_a e \).
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 \( \frac{1}{a} \log_a e \).
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