We are tasked with finding the limit:

\[
\lim_{{x \to 0}} \left( \frac{a^x + b^x + c^x}{3} \right)^{\frac{1}{x}}
\]

### Step-by-step analysis:

1. **Examine the base expression:**
   The expression inside the limit is:
   \[
   \left( \frac{a^x + b^x + c^x}{3} \right)^{\frac{1}{x}}
   \]
   As \( x \to 0 \), we note that for any positive real number \(a\), \(a^x\) approaches 1, because \(a^x = e^{x \ln a}\) and as \(x \to 0\), \(x \ln a \to 0\). Therefore:
   \[
   a^x \to 1, \quad b^x \to 1, \quad c^x \to 1 \quad \text{as} \quad x \to 0.
   \]

2. **Simplify the limit expression:**
   As \(x \to 0\), \(a^x + b^x + c^x \to 3\), so the base of the expression approaches:
   \[
   \frac{a^x + b^x + c^x}{3} \to \frac{1 + 1 + 1}{3} = 1.
   \]
   
3. **Apply the limit:**
   The expression then becomes:
   \[
   \left( \frac{a^x + b^x + c^x}{3} \right)^{\frac{1}{x}} \to 1^{\frac{1}{x}}.
   \]
   Since \(1^{\frac{1}{x}} = 1\) for any \(x\), the limit is simply:
   \[
   \lim_{{x \to 0}} 1 = 1.
   \]

Thus, the final result is:

\[
\boxed{1}.
\]

Would you like more details, or do you have any questions about this solution?

Here are 5 related questions for practice:
1. What is the limit \(\lim_{{x \to 0}} a^x\) for \(a > 0\)?
2. How do you compute \(\lim_{{x \to 0}} \left( \frac{a^x + b^x}{2} \right)^{1/x}\)?
3. What is the behavior of \(a^x\) as \(x \to \infty\) for \(a > 1\)?
4. What is the limit of \(\left( \frac{e^x + 1}{2} \right)^{1/x}\) as \(x \to 0\)?
5. How do you compute the limit \(\lim_{{x \to 0}} \left( \frac{a^x}{x} \right)\)?

**Tip:** When dealing with limits involving powers, it's helpful to rewrite expressions using logarithms for easier evaluation.

Explore the detailed step-by-step solution for finding the limit of ((a^x + b^x + c^x)/3)^1/x as x approaches 0. This problem involves exponential functions and properties of limits. Suitable for high school students in Grades 11-12.

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