Question

Evaluate the limit: \[ L = \lim_{x \to 0} \frac{35^x - 7^x - 5^x + 1}{(e^x - e^{-x}) \ln(1 - 3x)} \]

• A. \( \frac{\ln 35}{6} \)
• B. \( \frac{\ln 35}{6} \)
• C. \( \frac{\ln 2}{6} \)
• D. \( \frac{\ln (5) \cdot \ln 7}{-6} \)

Hint:

When evaluating limits involving exponentials and logarithms, use the small \( x \) approximations: \[ a^x - 1 \approx x \ln a, \quad e^x - e^{-x} \approx 2x, \quad \ln(1 - x) \approx -x. \]