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The question is about the rational function \( r \) and its asymptotic behavior based on the provided graph. Let's go through each part of the problem.

### Part (A): Asymptotes

#### (i) Vertical Asymptote
A **vertical asymptote** occurs where the function approaches infinity or negative infinity as \( x \) approaches a certain value, but the function itself is undefined at that point. In the graph, there is a vertical asymptote at \( x = 1 \) since the graph shoots off to infinity as \( x \) gets closer to 1 from both sides.

- **Equation of the vertical asymptote**: \( x = 1 \)

#### (ii) Horizontal Asymptote
A **horizontal asymptote** describes the behavior of the function as \( x \) approaches infinity or negative infinity. It represents the value that the function approaches as \( x \) increases or decreases without bound. In the graph, as \( x \) approaches either positive or negative infinity, the graph approaches \( y = -3 \).

- **Equation of the horizontal asymptote**: \( y = -3 \)

### Part (B): End Behavior

#### (i) End Behavior as \( x \to \infty \)
To describe the end behavior of the function \( r(x) \) as \( x \) increases without bound, we observe the graph as \( x \to \infty \). The graph gets closer and closer to the horizontal asymptote \( y = -3 \).

So, we can express this mathematically using limit notation:
\[
\lim_{x \to \infty} r(x) = -3
\]

#### (ii) End Behavior as \( x \to -\infty \)
Similarly, as \( x \to -\infty \), the graph also approaches the horizontal asymptote \( y = -3 \).

Thus, we write:
\[
\lim_{x \to -\infty} r(x) = -3
\]

### Final Answer Summary:
- (A) (i) The vertical asymptote of \( r \) is \( x = 1 \).
- (A) (ii) The horizontal asymptote of \( r \) is \( y = -3 \).
- (B) \( \lim_{x \to \infty} r(x) = -3 \) and \( \lim_{x \to -\infty} r(x) = -3 \).

Let me know if you'd like further clarification or additional details!

Here are some related questions:
1. How do you find the vertical asymptotes of a rational function algebraically?
2. Why does a rational function have a horizontal asymptote?
3. What role do the degrees of the numerator and denominator play in determining asymptotes?
4. How do you interpret the limit of a function at infinity?
5. Can a function cross a horizontal asymptote? Why or why not?

**Tip:** Vertical asymptotes occur where the denominator of a rational function is zero, but the numerator is non-zero at those points. Horizontal asymptotes are determined by the end behavior of the function as \( x \) approaches infinity or negative infinity.

The graph of the rational function r is shown above, along with the horizontal and vertical asymptotes of r. 
(A) (i) Write the equation of the line that is the vertical asymptote of r. 
(ii) Write the equation of the line that is the horizontal asymptote of r. 
(B) (i) Determine the end behavior of r as x increases without bound. Express your answer using the mathematical notation of a limit.

Learn how to determine the vertical and horizontal asymptotes of a rational function, and how to describe the end behavior using limit notation. The graph of the function shows asymptotic behavior at x = 1 and y = -3. This content is ideal for students in Grades 11-12.

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