Given the description of the graph, it seems that the function $A(x)$ has several points where the behavior is of interest. We need to analyze the limits of the function at specific points where the function is nearly vertical or exhibits a notable change. Here are the points of interest:

1. Limit as $x$ approaches $-3$ from the left ($x \to -3^-$):

• As the function approaches $x = -3$ from the left, it becomes nearly vertical and moves upwards.
• Limit: $\lim_{x \to -3^-} A(x) = \infty$

2. Limit as $x$ approaches $-3$ from the right ($x \to -3^+$):

• As the function reenters from the right of $x = -3$, it moves downwards and becomes less steep.
• Limit: $\lim_{x \to -3^+} A(x) = -\infty$

3. Limit as $x$ approaches $-1$ from the left ($x \to -1^-$):

• As the function approaches $x = -1$ from the left, it becomes nearly vertical and moves downwards.
• Limit: $\lim_{x \to -1^-} A(x) = -\infty$

4. Limit as $x$ approaches $-1$ from the right ($x \to -1^+$):

• As the function reenters from the right of $x = -1$, it moves upwards and becomes less steep.
• Limit: $\lim_{x \to -1^+} A(x) = \infty$

5. Limit as $x$ approaches $2$ from the left ($x \to 2^-$):

• As the function approaches $x = 2$ from the left, it becomes nearly vertical and moves downwards.
• Limit: $\lim_{x \to 2^-} A(x) = -\infty$

6. Limit as $x$ approaches $2$ from the right ($x \to 2^+$):

• As the function reenters from the right of $x = 2$, it moves upwards.
• Limit: $\lim_{x \to 2^+} A(x) = \infty$

7. Limit as $x$ approaches $\infty$ ($x \to \infty$):

• The function exits in the first quadrant, becoming more steep and heading upwards.
• Limit: $\lim_{x \to \infty} A(x) = \infty$

8. Limit as $x$ approaches $-\infty$ ($x \to -\infty$):

• The function exits in the second quadrant, becoming more steep and heading upwards.
• Limit: $\lim_{x \to -\infty} A(x) = \infty$

If you have any more questions or need further details, feel free to ask!

Here are 5 related questions:

1. How does the continuity of the function affect the limits at $x = -3$, $x = -1$, and $x = 2$?
2. What is the significance of the function changing direction at $x = 0.5$ and $x = 4.5$?
3. Can a function have both finite and infinite limits at the same point from different directions?
4. What is the definition of a vertical asymptote, and how does it relate to the behavior of the function near $x = -3$, $x = -1$, and $x = 2$?
5. How does the function's steepness impact the limit calculations?

Tip: Always check if a function approaches infinity or negative infinity as it nears certain points to identify possible vertical asymptotes.