Question

The limit $ \lim_{x \to 0} \frac{5^x + 4^x}{4^x - 3^x} $ is equal to

• A. 0
• B. \( \frac{\log(5/4)}{\log(4/3)} \)
• C. 1
• D. None of these

Hint:

When evaluating limits involving exponential functions, use approximations for small values of \( x \) to simplify the calculation.

Answer 1

The Correct Option is C

Step 1: Simplify the Expression
To compute this limit, use the fact that for small \( x \), \( a^x \approx 1 + x \log a \). Thus: \[ 5^x \approx 1 + x \log 5, \quad 4^x \approx 1 + x \log 4, \quad 3^x \approx 1 + x \log 3 \] Substituting these approximations into the given expression: \[ \frac{5^x + 4^x}{4^x - 3^x} \approx \frac{(1 + x \log 5) + (1 + x \log 4)}{(1 + x \log 4) - (1 + x \log 3)} = \frac{2 + x (\log 5 + \log 4)}{x (\log 4 - \log 3)} \]
Step 2: Evaluating the Limit
Taking the limit as \( x \to 0 \), the constant terms cancel out, and we are left with: \[ \lim_{x \to 0} \frac{2 + x (\log 5 + \log 4)}{x (\log 4 - \log 3)} = 1 \]
Step 3: Conclusion
Thus, the limit is equal to 1.