https://file.aiquickdraw.com/mathbot/file/1726351634047l4z91k2t.jpg

Let's analyze the graph and the questions presented:

1. **Limit as \( x \to \infty \) (Question a):**
   - We need to find \( \lim_{x \to \infty} f(x) \), meaning what happens to the function \( f(x) \) as \( x \) approaches positive infinity. From the graph, as \( x \) increases to large positive values, the function appears to approach the horizontal line \( y = 0 \), so the limit is 0.

2. **Limit as \( x \to -\infty \) (Question b):**
   - Similarly, for \( \lim_{x \to -\infty} f(x) \), as \( x \) becomes very negative, the function also seems to approach the horizontal line \( y = 0 \). Thus, the limit is 0.

3. **Limit as \( x \to 1 \) (Question c):**
   - Around \( x = 1 \), the graph shows a vertical asymptote. As \( x \) approaches 1 from either side, the function tends to \( \infty \) or \( -\infty \), indicating the function has no limit and tends to infinity.

4. **Limit as \( x \to 3 \) (Question d):**
   - At \( x = 3 \), the graph does not show any discontinuity, and the function seems to approach a specific value. Based on the graph, \( \lim_{x \to 3} f(x) \) appears to be about -1.

5. **Asymptotes (Question e):**
   - **Vertical asymptote**: From the graph, there is a vertical asymptote at \( x = 1 \).
   - **Horizontal asymptote**: The function approaches the horizontal line \( y = 0 \) as \( x \to \infty \) or \( x \to -\infty \), so the horizontal asymptote is \( y = 0 \).

### Answers:
- (a) \( \lim_{x \to \infty} f(x) = 0 \)
- (b) \( \lim_{x \to -\infty} f(x) = 0 \)
- (c) \( \lim_{x \to 1} f(x) = \infty \)
- (d) \( \lim_{x \to 3} f(x) = -1 \)
- (e) Vertical asymptote: \( x = 1 \), Horizontal asymptote: \( y = 0 \)

Would you like further details on any part? Here are five questions that can deepen your understanding:

1. Why does the function have no limit at \( x = 1 \)?
2. What is the difference between vertical and horizontal asymptotes?
3. How can you tell if a graph approaches a specific value as \( x \to \infty \)?
4. Why does the limit at \( x = 3 \) exist and equal -1, unlike at \( x = 1 \)?
5. What are common behaviors of functions around vertical asymptotes?

**Tip:** When determining limits graphically, watch for asymptotes and the general trend of the function's behavior.

For the function f whose graph is given, state the following: (a) lim x→∞ f(x), (b) lim x→−∞ f(x), (c) lim x→1 f(x), (d) lim x→3 f(x), (e) the equations of the asymptotes (vertical and horizontal).

This problem involves analyzing a function's graph to calculate limits as x approaches infinity, negative infinity, and specific values. It also includes identifying the equations of vertical and horizontal asymptotes. Suitable for high school calculus students.

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