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Let's go through the questions using the given graph of the function \( f \):

### (a) Finding Limits

1. **\(\lim_{x \to 2^-} f(x)\):**
   - As \( x \) approaches 2 from the left, \( f(x) \) approaches a vertical asymptote, heading towards \( +\infty \).

2. **\(\lim_{x \to 2^+} f(x)\):**
   - As \( x \) approaches 2 from the right, \( f(x) \) also heads towards a vertical asymptote, but this time going towards \( -\infty \).

3. **\(\lim_{x \to 3} f(x)\):**
   - As \( x \) approaches 3, the function seems to approach a finite value. By observing the graph, the limit appears to be \( 1 \).

4. **\(\lim_{x \to 4^-} f(x)\):**
   - As \( x \) approaches 4 from the left, \( f(x) \) approaches the value at the open circle, which seems to be at \( y = 2 \).

5. **\(\lim_{x \to 0} f(x)\):**
   - As \( x \) approaches 0, \( f(x) \) approaches \( 0 \) (since \( f(0) = 0 \)).

6. **\(\lim_{x \to -3} f(x)\):**
   - As \( x \) approaches \(-3\), the function seems to head towards a vertical asymptote at \( y \to -\infty \).

7. **\(\lim_{x \to \infty} f(x)\):**
   - As \( x \) goes to infinity, the function levels off towards a horizontal asymptote, approaching \( y = 2 \).

8. **\(\lim_{x \to -\infty} f(x)\):**
   - As \( x \) goes to negative infinity, the function levels off towards a horizontal asymptote at \( y = 0 \).

### (b) Horizontal Asymptotes
- From the graph:
  1. \( y = 0 \) as \( x \to -\infty \)
  2. \( y = 2 \) as \( x \to \infty \)

### (c) Vertical Asymptotes
- The function has vertical asymptotes at:
  1. \( x = -3 \)
  2. \( x = 2 \)

### (d) Discontinuities
- The function is discontinuous at the following points:
  1. \( x = -3 \) (vertical asymptote)
  2. \( x = 2 \) (vertical asymptote)
  3. \( x = 4 \) (jump discontinuity where the function has a hole at \( x = 4 \) and a different point value elsewhere)

Would you like more detailed explanations for any of these limits or concepts? 

### Relative Questions
1. What are the types of discontinuities in a function and how are they identified?
2. How do you determine the behavior of a function as it approaches a vertical asymptote?
3. How are horizontal asymptotes different from slant asymptotes, and when do slant asymptotes occur?
4. How can the limit at a point help in determining the continuity of a function?
5. What role do asymptotes play in understanding the end behavior of a function?

### Tip
When analyzing a graph for limits, focus on the direction from which \( x \) approaches a point, as limits can differ from the left-hand side and right-hand side.

Find the limits and asymptotes of the given graph of f(x) and determine discontinuities.

This problem involves analyzing the graph of a function to determine its limits, horizontal and vertical asymptotes, and points of discontinuity. It demonstrates the behavior of a function near critical points and asymptotes. Ideal for Grades 10-12, focusing on key calculus concepts.

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