Question

\[ \lim_{x \to 0} \frac{a^x - b^x}{x} \text{ is equal to:} \]

• A. \( \log ab \)
• B. \( \log b \)
• C. \( \log \frac{a}{b} \)
• D. \( \log a \)

Hint:

For limits of the form \( \frac{a^x - b^x}{x} \) as \( x \to 0 \), use the approximation \( c^x \approx 1 + x \ln c \) to simplify the expression.

Answer 1

The Correct Option is C

We are asked to find the limit: \[ \lim_{x \to 0} \frac{a^x - b^x}{x} \] Step 1: Use the logarithmic approximation for small \( x \). For any constant \( c \), we know the approximation for \( c^x \) when \( x \) is near 0: \[ c^x \approx 1 + x \ln c \] Using this approximation for \( a^x \) and \( b^x \), we get: \[ a^x \approx 1 + x \ln a \] \[ b^x \approx 1 + x \ln b \] Step 2: Substitute into the limit. Substitute these approximations into the original expression: \[ \frac{a^x - b^x}{x} \approx \frac{(1 + x \ln a) - (1 + x \ln b)}{x} \] Simplifying: \[ = \frac{x (\ln a - \ln b)}{x} \] The \( x \)'s cancel out: \[ = \ln a - \ln b \] This simplifies to: \[ = \log \frac{a}{b} \] Thus, the value of the limit is \( \log \frac{a}{b} \). Therefore, the correct answer is option (C)