https://file.aiquickdraw.com/mathbot/file/1727162563155bnl8n9q3.jpg

Let's analyze the graph and solve the limits as \( x \to 3 \) for the function \( f(x) \):

### 1. \(\lim_{x \to 3^-} f(x)\)
This is the left-hand limit, meaning we approach \( x = 3 \) from the left (values less than 3). From the graph, as \( x \to 3^- \), the function \( f(x) \) approaches \(-1\).

\[
\lim_{x \to 3^-} f(x) = -1
\]

### 2. \(\lim_{x \to 3^+} f(x)\)
This is the right-hand limit, meaning we approach \( x = 3 \) from the right (values greater than 3). From the graph, as \( x \to 3^+ \), the function approaches \(-2\), as shown by the open circle at \( (3, -2) \).

\[
\lim_{x \to 3^+} f(x) = -2
\]

### 3. \(\lim_{x \to 3} f(x)\)
The limit at \( x = 3 \) only exists if the left-hand and right-hand limits are equal. Since:
\[
\lim_{x \to 3^-} f(x) = -1 \quad \text{and} \quad \lim_{x \to 3^+} f(x) = -2
\]
the left-hand and right-hand limits are not equal. Therefore, the overall limit does not exist.

\[
\lim_{x \to 3} f(x) = \text{DNE}
\]

### Summary of Results:
1. \(\lim_{x \to 3^-} f(x) = -1\)
2. \(\lim_{x \to 3^+} f(x) = -2\)
3. \(\lim_{x \to 3} f(x) = \text{DNE}\)

Would you like more details on any of these steps?

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Here are 5 related questions:
1. What are the conditions for a limit to exist at a point?
2. How can you interpret a discontinuity from a graph?
3. What does it mean if a limit tends to infinity?
4. Can a function be continuous but not differentiable at a point? 
5. How do one-sided limits help in understanding step functions?

**Tip:** When a limit does not exist, it is often because the function has different behaviors from the left and right sides.

Determine the following values based on the graph of f(x): 
1. lim_{x→3^-} f(x) 
2. lim_{x→3^+} f(x) 
3. lim_{x→3} f(x).

Learn how to determine the one-sided and overall limits of a function from a graph at x = 3. This problem covers key concepts in limits and discontinuities, perfect for students in Grades 11-12.

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