Question

The value of
\[ \lim_{x \to \infty} \left(a_1^{1/x} + a_2^{1/x} + \cdots + a_n^{1/x}\right)^{n x}, \quad a_i > 0 \]

• A. \(a_1+a_2+\cdots+a_n\)
• B. \(e^{a_1+a_2+\cdots+a_n}\)
• C. \(\dfrac{a_1+a_2+\cdots+a_n}{n}\)
• D. a₁a₂⋯ aₙ

Hint:

Use logarithms to evaluate limits of powers.

Answer 1

The Correct Option is D

As \(x \to \infty\),
\[ a_i^{1/x} \approx 1 + \frac{\ln a_i}{x}. \]
Taking logarithm and limit gives:
\[ \ln L = \sum_{i=1}^{n} \ln a_i \implies L = a_1 a_2 \cdots a_n. \]