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 $\lim_{x \to -\infty} r(x) = -3$.

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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.