We need to evaluate the limit:

$\lim_{x \to \infty} \left( 2^x + 3^x + 5^x \right)^{\frac{1}{x}}.$

To approach this, let's consider the asymptotic behavior of each term as $x \to \infty$.

Step 1: Dominant Term

As $x \to \infty$, the term $5^x$ dominates the terms $2^x$ and $3^x$ because $5^x$ grows exponentially faster than $2^x$ and $3^x$. Therefore, for large $x$, we can approximate the expression:

$2^x + 3^x + 5^x \approx 5^x.$

Step 2: Simplify the Expression

Using this approximation, we can simplify the expression inside the limit:

$\left( 2^x + 3^x + 5^x \right)^{\frac{1}{x}} \approx \left( 5^x \right)^{\frac{1}{x}}.$

Step 3: Evaluate the Limit

Now, we evaluate the simplified expression:

$\left( 5^x \right)^{\frac{1}{x}} = 5.$

Thus, the limit is:

$\lim_{x \to \infty} \left( 2^x + 3^x + 5^x \right)^{\frac{1}{x}} = 5.$

Conclusion:

The value of the limit is $\boxed{5}$.

Would you like further details on any part of this solution? Here are some related questions you might find helpful:

1. What is the general approach for limits involving exponential functions?
2. Why does $5^x$ dominate over $2^x$ and $3^x$ as $x \to \infty$?
3. How would the result change if the base were different (e.g., $6^x$ instead of $5^x$)?
4. What is the limit of $(a^x + b^x)^{1/x}$ for any $a$ and $b$?
5. What other types of functions grow faster than exponential functions?

Tip: When dealing with limits involving multiple exponential terms, always identify the dominant term, which grows the fastest as $x \to \infty$. This often simplifies the limit significantly.