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.