https://r2.erweima.ai/mathbot/image/1725998729998s1wxygc2.jpg

Let's analyze the graph step by step and go through each of the limits and the vertical asymptote questions in the image.

### (a) \(\lim_{x \to -3} A(x)\)
- At \(x = -3\), the graph shows that the function seems to approach \(+\infty\) from the left and right. This indicates there is a vertical asymptote at \(x = -3\), so the limit does not exist (DNE).

### (b) \(\lim_{x \to 2^-} A(x)\)
- Approaching \(x = 2\) from the left side, the graph shows the function goes to negative infinity, \(-\infty\). Therefore, the correct limit is \(-\infty\).

### (c) \(\lim_{x \to 2^+} A(x)\)
- Approaching \(x = 2\) from the right side, the graph indicates that the function rises towards infinity. Thus, the correct limit is \(+\infty\), which was marked correctly.

### (d) \(\lim_{x \to -1} A(x)\)
- As \(x\) approaches \(-1\), the graph smoothly approaches a value close to \(1\), not zero. So the correct answer here should be \(1\), not \(0\).

### (e) Vertical Asymptotes
- Vertical asymptotes occur where the function heads to infinity or negative infinity on both sides of a given \(x\)-value. Based on the graph, there are vertical asymptotes at \(x = -3\) and \(x = 2\).

#### Corrected Answers:
1. (a) DNE (since the function approaches infinity at \(x = -3\)).
2. (b) \(-\infty\).
3. (c) \(+\infty\) (this was already correct).
4. (d) The limit is \(1\), not \(0\).
5. (e) Vertical asymptotes at \(x = -3\) and \(x = 2\).

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Do you need further clarification or details on any of these answers?

Here are some related questions to deepen your understanding:
1. What is the definition of a vertical asymptote, and how do you identify it on a graph?
2. How do left-hand and right-hand limits differ, and why are they useful in determining limits at a point?
3. Why does the limit not exist at \(x = -3\), and what are the implications for the function?
4. How can you tell if a function approaches infinity or negative infinity near an asymptote?
5. What strategies can you use to find limits algebraically when a graph is not available?

**Tip:** Always examine the behavior of the function on both sides of a point when finding limits. This is crucial, especially near vertical asymptotes.

For the function A whose graph is shown, state the following limits and vertical asymptotes. If the limit does not exist, enter DNE.

Analyze the graph of function A to determine limits at specified points and locate vertical asymptotes. This problem involves the concepts of limits approaching infinity, asymptotes, and the behavior of functions near undefined points.

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