https://file.aiquickdraw.com/mathbot/file/1727229678139uhzuigx7.jpg

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.

Analyze the function f(x) and answer questions about the limits and function values when x = -3 based on the graph provided.

Learn how to analyze the left-hand and right-hand limits, and the value of the function f(x) at x = -3. Understand how a jump discontinuity affects the overall limit.

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