The problem given in the second image is:

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

Steps to Solve:

1. Behavior of $a^x$, $b^x$, and $c^x$ as $x \to 0$:

   • For any positive number $a$, $a^x$ approaches 1 as $x \to 0$.
   • Therefore, as $x$ approaches 0, the expression $a^x + b^x + c^x$ approaches $1 + 1 + 1 = 3$.

2. Simplification: The expression inside the limit simplifies to: $\lim_{{x \to 0}} \left(\frac{a^x + b^x + c^x}{3}\right)^{\frac{1}{x}} = \lim_{{x \to 0}} \left(\frac{3}{3}\right)^{\frac{1}{x}} = \lim_{{x \to 0}} 1^{\frac{1}{x}}.$ Since $1^{\frac{1}{x}} = 1$ for all $x$, this expression equals 1.

Final Answer:

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

This result holds regardless of the values of $a$, $b$, and $c$, as long as they are positive.

Do you need any more details, or have any other questions?

Here are 5 related questions to explore further:

1. How would the limit change if $x \to 0^-$ (approaching from the negative side)?
2. What happens if $a$, $b$, and $c$ are not positive?
3. How would this limit be affected if we had an additional term in the sum?
4. Can you generalize this limit for a sum of more than three terms?
5. What happens if instead of $\frac{1}{x}$, the exponent was $x$?

Tip: When taking limits as $x \to 0$, remember to consider the behavior of the function at 0, especially for exponential and logarithmic expressions.