https://file.aiquickdraw.com/mathbot/file/1729433250638h634l1vz.jpg

Let me go through each part of the image to help you solve the questions presented:

### Part 1 (a)
- **Question**: As \( x \to \infty \), \( f(x) \to 2 \). 
  - **Answer**: The function is approaching the value 2 as \( x \) becomes infinitely large, which means there is a horizontal asymptote at \( y = 2 \).

### Part 1 (b)
- **Question**: As \( x \to -\infty \), \( f(x) \to -\infty \).
  - **Answer**: As \( x \) approaches negative infinity, \( f(x) \) is unbounded, heading towards negative infinity.

### Part 2 (c)
- **Question**: As \( x \to 2^+ \), \( f(x) \to \infty \).
  - **Answer**: As \( x \) approaches 2 from the right-hand side, the function approaches positive infinity, indicating a vertical asymptote at \( x = 2 \).

### Part 3 (d)
- **Question**: As \( x \to \infty \), \( f(x) \to 1 \).
  - **Answer**: As \( x \) approaches positive infinity, the function approaches a value of 1, meaning there is a horizontal asymptote at \( y = 1 \).

### Part 4
- **Graph Behavior**:
  - **Is the graph increasing over at least one interval?** Yes.
  - **Is the graph decreasing over at least one interval?** Yes.

### Part 5 (e)
- **Question**: The graph is decreasing over the interval(s). Provide the interval(s) in interval notation.
  - **Answer**: Without seeing the graph itself, you would need to determine from the function or provided graph where it decreases. Based on the information we have, the answer should be entered in proper interval notation based on the graph's behavior.

---

Would you like further clarifications or more details on any specific part? Here are five related questions for expanding the information:

1. How do you find horizontal asymptotes from a given function?
2. What is the significance of vertical asymptotes in the behavior of functions?
3. How can you determine whether a graph is increasing or decreasing from the derivative of the function?
4. What methods can you use to express intervals of increase or decrease in interval notation?
5. How does one approach finding limits at infinity?

**Tip**: For interval notation, always use parentheses for open intervals (where endpoints are not included) and brackets for closed intervals (where endpoints are included).

The original question submitted by the user involves limits, asymptotic behavior, and graph analysis.

This problem covers limits at infinity, horizontal and vertical asymptotes, and how to determine increasing and decreasing intervals on a graph. It involves concepts of calculus and precalculus, suitable for high school 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