For the given graph and question, I will now focus on the case where $a = 3$, which you've requested.

We are asked to determine the following:

1. $\lim\limits_{x \to 3^-} f(x)$ (the left-hand limit as $x$ approaches 3)
2. $\lim\limits_{x \to 3^+} f(x)$ (the right-hand limit as $x$ approaches 3)
3. $\lim\limits_{x \to 3} f(x)$ (the overall limit as $x$ approaches 3)
4. $f(3)$, if it exists (the function value at $x = 3$).

Analysis for $a = 3$:

1. Left-hand limit $\lim\limits_{x \to 3^-} f(x)$:

• From the graph, as $x$ approaches 3 from the left, the function value follows a line heading towards the point $(3, -2)$.
• So, $\lim\limits_{x \to 3^-} f(x) = -2$.

2. Right-hand limit $\lim\limits_{x \to 3^+} f(x)$:

• As $x$ approaches 3 from the right, the function value follows a line heading towards the point $(3, 1)$.
• Thus, $\lim\limits_{x \to 3^+} f(x) = 1$.

3. Overall limit $\lim\limits_{x \to 3} f(x)$:

• Since the left-hand limit and the right-hand limit are not equal (i.e., $-2 \neq 1$), the overall limit at $x = 3$ does not exist.

4. Function value $f(3)$:

• From the graph, there is an open circle at both points $(3, -2)$ and $(3, 1)$, indicating that the function is not defined at $x = 3$.
• Therefore, $f(3)$ does not exist.

Summary for $a = 3$:

• $\lim\limits_{x \to 3^-} f(x) = -2$
• $\lim\limits_{x \to 3^+} f(x) = 1$
• $\lim\limits_{x \to 3} f(x)$ does not exist
• $f(3)$ does not exist

Do you need more details or explanations on any part?

Related Questions:

1. What is the definition of a limit from the left and the right in terms of continuity?
2. How can discontinuities be classified based on limits and function values?
3. What conditions must be met for the overall limit to exist?
4. How would the graph look if the function were continuous at $x = 3$?
5. How do we interpret open circles in graphs with respect to function values?

Tip:

For a limit to exist at a point, the left-hand and right-hand limits must be equal, and the function doesn't have to be defined at that point.