Question

The value of \(\lim_{x \to \infty} \left\{ \frac{a_1^{1/x} + a_2^{1/x} + .......... + a_n^{1/x}}{n} \right\}^x\) is

• A. \(a_1 + a_2 + .......... + a_n\)
• B. \(e^{a_1 + a_2 + .......... + a_n}\)
• C. \(\frac{a_1 + a_2 + .......... + a_n}{n}\)
• D. \(a_1 a_2 .......... a_n\)

Hint:

This is a standard limit: \(\lim_{x \to 0} \left( \frac{\sum a_i^x}{n} \right)^{1/x} = \left( \prod a_i \right)^{1/n}\).

Answer 1

The Correct Option is D

To find the value of the given limit, we need to analyze the expression:

\[\lim_{x \to \infty} \left( \frac{a_1^{1/x} + a_2^{1/x} + \ldots + a_n^{1/x}}{n} \right)^x\]

We will solve this using the properties of limits and exponentials:

1. Observe that \(a_i^{1/x}\) approaches 1 as \(x\) goes to infinity for each \(i\). 
2. Thus, the sum in the numerator becomes:
3. Therefore, the overall expression inside the limit simplifies to:
4. Now we focus on the term raised to the power of \(x\):
5. However, using the exponential limit property, if we rewrite the exponent:
   For large \(x\), \(a_i^{1/x} \approx 1 + \frac{\ln a_i}{x}\)
6. Substituting this, the original expression becomes:
7. Simplified as:
8. Using the limit property, \(\left(1 + \frac{k}{x}\right)^x \to e^k\) as \(x \to \infty\), we get:

Therefore, the value of the limit is:

\(a_1 a_2 \ldots a_n\)

This matches with the given correct answer option:

\(a_1 a_2 \ldots a_n\)