lim x -> infinity (2 ^ x + 3 ^ x + 5 ^ x) ^ (1/x)

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.

Explore how to evaluate the limit lim x -> infinity (2^x + 3^x + 5^x)^(1/x). This solution uses the concept of exponential growth dominance and simplification techniques. Ideal for high school and early college students studying limits and exponential functions.

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