Let's break this problem down and analyze the discontinuity in the function graph.

Step 1: Identify the point of discontinuity.

Looking at the graph description, the function seems to be discontinuous at x = -3. At this point, there is a jump in the function values. Specifically, the graph has:

• An open point at $(-3, -1)$, indicating that the function is not defined at $x = -3$ with the value $-1$.
• A closed point at $(-3, 9)$, showing that the function has the value 9 at $x = -3$.

Therefore, $a = -3$ is the value where the function is discontinuous.

Step 2: Analyze the function behavior at $x = a = -3$.

(a) $f(a)$ if it exists:

The closed point at $(-3, 9)$ indicates that $f(-3) = 9$.

(b) $\lim_{{x \to a^-}} f(x)$:

The left-hand limit means the value of the function as $x$ approaches $-3$ from the left side. From the description, as $x$ approaches $-3$ from the left, the function approaches $-1$, due to the open point at $(-3, -1)$. Hence, $\lim_{{x \to -3^-}} f(x) = -1.$

(c) $\lim_{{x \to a^+}} f(x)$:

The right-hand limit means the value of the function as $x$ approaches $-3$ from the right side. Based on the graph, the function also approaches $-1$ from the right side. Hence, $\lim_{{x \to -3^+}} f(x) = -1.$

(d) $\lim_{{x \to a}} f(x)$:

The overall limit as $x$ approaches $-3$ only exists if the left-hand and right-hand limits are equal. Since both limits are equal to $-1$, we have: $\lim_{{x \to -3}} f(x) = -1.$

(e) Which conditions for continuity are not met?

For a function to be continuous at a point $x = a$, three conditions must be satisfied:

1. $f(a)$ must exist.
2. $\lim_{{x \to a}} f(x)$ must exist.
3. $f(a) = \lim_{{x \to a}} f(x)$.

In this case:

• $f(-3) = 9$,
• $\lim_{{x \to -3}} f(x) = -1$.

Since $f(-3) \neq \lim_{{x \to -3}} f(x)$, the function is discontinuous at $x = -3$. The condition that is not met is that $f(a)$ does not equal the limit of the function as $x$ approaches $a$.

Summary:

• $a = -3$,
• $f(-3) = 9$,
• $\lim_{{x \to -3^-}} f(x) = -1$,
• $\lim_{{x \to -3^+}} f(x) = -1$,
• $\lim_{{x \to -3}} f(x) = -1$,
• Discontinuity occurs because $f(-3) \neq \lim_{{x \to -3}} f(x)$.

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Would you like to dive deeper into any of these details?

Here are 5 related questions to consider:

1. What are the different types of discontinuities a function can have?
2. How do you determine if a function is continuous at a point?
3. What is the significance of left-hand and right-hand limits?
4. How can a piecewise function be written for this problem?
5. What are the conditions for a function to have a removable discontinuity?

Tip: When analyzing discontinuities, always check if the left-hand and right-hand limits are equal to identify the type of discontinuity.