Question

The value of \( \lim_{x \to 0} \left( \frac{a^x + b^x + c^x}{3} \right)^{\frac{2}{x}} \), \( (a,b,c>0) \) is

• A. \( (abc)^3 \)
• B. \( abc \)
• C. \( (abc)^{1/3} \)
• D. None of these

Hint:

For limits involving powers, convert to exponential form using \( (1+x)^n \to e^{nx} \).

Answer 1

The Correct Option is D

Concept: Use logarithm and exponential expansion: \[ a^x = 1 + x\ln a + o(x) \]

Step 1: Expand terms using approximation. \[ a^x + b^x + c^x = 3 + x(\ln a + \ln b + \ln c) \] \[ = 3 + x \ln(abc) \]

Step 2: Divide by 3: \[ \frac{a^x + b^x + c^x}{3} = 1 + \frac{x}{3}\ln(abc) \]

Step 3: Apply limit: \[ \left(1 + \frac{x}{3}\ln(abc)\right)^{\frac{2}{x}} \] \[ = e^{\frac{2}{3}\ln(abc)} = (abc)^{\frac{2}{3}} \] Final Answer: \[ (abc)^{\frac{2}{3}} \notin \text{options} \]