Let's analyze the graph to answer the questions about the function $f(x)$ when $x = -3$.

1. Left-hand limit as $x$ approaches -3: We need to observe the value that $f(x)$ approaches as $x$ approaches -3 from the left (i.e., as $x \to -3^{-}$).

   • As $x$ approaches -3 from the left, the curve approaches a value of 4. Thus, $\lim_{x \to -3^-} f(x) = 4$.

2. Right-hand limit as $x$ approaches -3: Next, we observe the behavior of $f(x)$ as $x$ approaches -3 from the right (i.e., as $x \to -3^{+}$).

   • As $x \to -3^+$, the graph shows that $f(x)$ approaches 2. Therefore, $\lim_{x \to -3^+} f(x) = 2$.

3. Overall limit as ( x \to -3: The overall limit exists only if both the left-hand and right-hand limits are equal. Since $\lim_{x \to -3^-} f(x) = 4$ and $\lim_{x \to -3^+} f(x) = 2$, the two limits are not equal. Therefore, the overall limit does not exist (DNE).

4. The value of the function at $x = -3$: From the graph, we can directly observe the value of $f(x)$ at $x = -3$. There is a filled dot on the graph at $x = -3$ with a corresponding $y$-value of 3. Hence, $f(-3) = 3$.

Final answers:

$\lim_{x \to -3^-} f(x) = 4$ $\lim_{x \to -3^+} f(x) = 2$ $\lim_{x \to -3} f(x) = \text{DNE}$ $f(-3) = 3$

Would you like more details on any step or further clarification?

Here are five relative questions:

1. What happens to the limit if the left-hand and right-hand limits are equal?
2. How do discontinuities affect the limits of a function?
3. What are the differences between limits and actual function values?
4. Can a function be continuous but have a limit that does not exist?
5. How can we calculate limits for more complex functions analytically?

Tip: When a function has a jump discontinuity (as seen here at $x = -3$), the overall limit does not exist due to the mismatch between the left-hand and right-hand limits.