https://file.aiquickdraw.com/mathbot/file/1726284466136yjtag8uk.jpg

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:

\[
\text{a. } -24
\]

Would you like further details or explanations?

### 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.

Misalkan lim(x→c)f(x)=2 dan lim(x→c)g(x)=−2. Nilai dari lim(x→c)(f^3(x) + g^5(x)) adalah ...

This problem involves calculating the limit of a sum of powers of functions. Learn how to apply the rules of limits to find the solution. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation