$\lim _{x \rightarrow 0}\left(\frac{a^{x}+b^{x}+c^{x}}{3}\right)^{\frac{1}{x}}$

Step 1 ：Find the limit of the given expression as x approaches 0: $\lim _{x \rightarrow 0}\left(\frac{a^{x}+b^{x}+c^{x}}{3}\right)^{\frac{1}{x}}$

Step 2 ：Find the derivative of the numerator and the denominator with respect to x:

Step 3 ：numerator = $\frac{a^x}{3} + \frac{b^x}{3} + \frac{c^x}{3}$

Step 4 ：denominator = x

Step 5 ：numerator_derivative = $\frac{a^x \log(a)}{3} + \frac{b^x \log(b)}{3} + \frac{c^x \log(c)}{3}$

Step 6 ：denominator_derivative = 1

Step 7 ：Apply L'Hopital's rule to find the limit of the expression as x approaches 0:

Step 8 ：limit_expression = $\frac{a^x \log(a)}{3} + \frac{b^x \log(b)}{3} + \frac{c^x \log(c)}{3}$

Step 9 ：limit_result = $\frac{\log(a)}{3} + \frac{\log(b)}{3} + \frac{\log(c)}{3}$

Step 10 ：Final Answer: $\boxed{\frac{\log(a) + \log(b) + \log(c)}{3}}$