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?

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.