The problem states:

Given: $\lim_{x \to c} f(x) = 2 \quad \text{and} \quad \lim_{x \to c} g(x) = -2,$ find the value of: $\lim_{x \to c} \left(f^3(x) + g^5(x)\right).$

Step-by-Step Solution

1. Calculate $\lim_{x \to c} f^3(x)$: Since $\lim_{x \to c} f(x) = 2$, we have: $\lim_{x \to c} f^3(x) = \left(\lim_{x \to c} f(x)\right)^3 = 2^3 = 8.$

2. Calculate $\lim_{x \to c} g^5(x)$: Since $\lim_{x \to c} g(x) = -2$, we have: $\lim_{x \to c} g^5(x) = \left(\lim_{x \to c} g(x)\right)^5 = (-2)^5 = -32.$

3. Combine the limits: $\lim_{x \to c} \left(f^3(x) + g^5(x)\right) = \lim_{x \to c} f^3(x) + \lim_{x \to c} g^5(x) = 8 + (-32) = -24.$

Thus, the answer is:

$\text{a. } -24$

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Additional Related Questions

1. What is the general rule for the limit of a sum of functions?
2. How do you find the limit of a function raised to a power?
3. What is the difference between finding the limit of a product and the limit of a sum of functions?
4. How do limits behave when involving negative numbers and powers?
5. How would the solution change if the limits of $f(x)$ and $g(x)$ were different?

Tip: Remember that limits can be distributed over addition and constant powers if they exist.