Question

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

Answer 1

What are the \[ a_i's\]

Answer 2

It's not given.

Answer 3

But after seeing your previous solution, I think I can do it on my own now.
\[\text{AM}\ge \text{GM}\]\[\large \frac{a_1^{1/x}+a_2^{1/x}+\ldots+a_n^{1/x}}n \ge \left(a_1^{1/x} a _2^{1/x}a_3^{1/x}\ldots a_n^{1/x}\right)^{1/n}\]\[\large\frac{a_1^{1/x}+a_2^{1/x}+\ldots+a_n^{1/x}}n \ge \left(a_1 a _2a_3\ldots a_n\right)^{1/xn}\]
\[\large\left(\frac{a_1^{1/x}+a_2^{1/x}+\ldots+a_n^{1/x}}n\right)^{xn} \ge a_1 a _2a_3\ldots a_n\]Is it right?