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Let's analyze the problem and graph you provided.

### (a) Limits at specific points
The problem asks you to find the limit of \( f(x) \) as \( x \) approaches the following values: \( \alpha = -3 \), \( \alpha = 0 \), and \( \alpha = 1 \). If the limit doesn't exist, explain why.

#### 1. \( \alpha = -3 \)
- The function has a solid point at \( (-3, 2) \), which indicates that the function is continuous there.
- Since the graph shows no jumps or gaps near \( x = -3 \), the limit of \( f(x) \) as \( x \to -3 \) is simply the value of the function at \( x = -3 \), which is:
  \[
  \lim_{x \to -3} f(x) = 2
  \]

#### 2. \( \alpha = 0 \)
- The graph has an open circle at \( (0, -2) \), meaning the function is not defined at \( x = 0 \), but the curve approaches \( y = -2 \) from both sides.
- The left-hand limit and right-hand limit both approach \( -2 \), so:
  \[
  \lim_{x \to 0} f(x) = -2
  \]

#### 3. \( \alpha = 1 \)
- At \( x = 1 \), there is a jump discontinuity. The left-hand limit as \( x \to 1^- \) approaches \( 2 \), but the right-hand limit as \( x \to 1^+ \) approaches \( -3 \).
- Since the left-hand and right-hand limits are not equal, the limit does not exist:
  \[
  \lim_{x \to 1} f(x) \text{ does not exist.}
  \]

### (b) Intervals of Continuity
Continuity means the graph has no breaks, jumps, or holes within a given interval. Based on the graph:

- The function is continuous from \( x = -5 \) to \( x = -3 \), including \( -5 \) and \( -3 \) because the points are solid.
- It's continuous again from \( x = -3 \) to \( x = 0 \), excluding \( x = 0 \) because of the open circle.
- It's continuous from \( x = 0 \) to \( x = 1 \), excluding \( x = 1 \) due to the jump discontinuity.
- Finally, it is continuous from \( x = 1 \) to \( x = 5 \), excluding \( x = 5 \) because of the open circle at \( (5, 3) \).

So, the intervals of continuity are:
\[
[-5, -3] \cup (-3, 0) \cup (0, 1) \cup (1, 5)
\]

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Would you like further details on any part of the solution?

Here are 5 related questions for practice:
1. What is the definition of a limit at a point?
2. How do you find one-sided limits from a graph?
3. What are the conditions for continuity of a function at a point?
4. Can a function be continuous but have a removable discontinuity? Explain.
5. How do you determine whether a limit exists from both sides of a point?

**Tip:** Always check both one-sided limits when the graph shows a potential discontinuity, as this will help confirm whether the two-sided limit exists.

For the graph provided, (a) Find the limit of f(x) as x approaches -3, 0, and 1. (b) List the intervals of continuity for the function.

Analyze the limits of the function as x approaches specific values and determine the intervals of continuity based on the provided graph. This problem involves the concepts of limits and continuity, suitable for Grades 11-12.

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