The given limit is:

$\lim_{x \to \infty} \left( a_1^x + a_2^x + \cdots + a_n^x \right)^{\frac{1}{x}}$

Step 1: Factor Out the Maximum Term

Assume that $a_1, a_2, \dots, a_n$ are positive real numbers. Let $a_{\max} = \max(a_1, a_2, \dots, a_n)$, i.e., the largest value among them.

For large $x$, the term $a_{\max}^x$ dominates all other terms, meaning:

$a_1^x + a_2^x + \cdots + a_n^x \approx a_{\max}^x$

Step 2: Compute the Limit

Using this approximation, we can rewrite the expression inside the limit as:

$\left( a_{\max}^x \left( 1 + \sum_{\substack{i=1 \\ i \neq \max}}^n \left( \frac{a_i}{a_{\max}} \right)^x \right) \right)^{\frac{1}{x}}$

Since all terms $\left( \frac{a_i}{a_{\max}} \right)^x$ tend to zero as $x \to \infty$ (for $a_i < a_{\max}$), we obtain:

$\left( a_{\max}^x \cdot 1 \right)^{\frac{1}{x}} = a_{\max}$

Conclusion:

$\lim_{x \to \infty} \left( a_1^x + a_2^x + \cdots + a_n^x \right)^{\frac{1}{x}} = a_{\max}$

Thus, the limit evaluates to the maximum among $a_1, a_2, \dots, a_n$.

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Would you like a more detailed breakdown? Here are some related questions:

1. What happens if some $a_i$ are negative?
2. How does the result change if $x$ approaches $-\infty$ instead?
3. Can we generalize this result to an integral form?
4. What is an alternative proof using logarithms?
5. How does this relate to the concept of power means?

Tip: This type of limit is useful in asymptotic analysis, particularly in algorithms and optimization problems!