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: $[-5, -3] \cup (-3, 0) \cup (0, 1) \cup (1, 5)$

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.