Let's analyze the graph and determine the one-sided and two-sided limits:

1. $\lim\limits_{x \to 1^-} f(x)$

This is the limit of $f(x)$ as $x$ approaches 1 from the left. As $x \to 1^-$, the value of $f(x)$ approaches -2.

$\lim\limits_{x \to 1^-} f(x) = -2$

2. $\lim\limits_{x \to 1^+} f(x)$

This is the limit of $f(x)$ as $x$ approaches 1 from the right. As $x \to 1^+$, the value of $f(x)$ also approaches -2.

$\lim\limits_{x \to 1^+} f(x) = -2$

3. $\lim\limits_{x \to 1} f(x)$

Since both the left-hand limit and right-hand limit at $x = 1$ are equal, the two-sided limit exists and is equal to -2.

$\lim\limits_{x \to 1} f(x) = -2$

4. $\lim\limits_{x \to 2} f(x)$

At $x = 2$, the function has a hole, meaning the function does not exist at this point. However, the value of the function from both the left and right approaches -1.

$\lim\limits_{x \to 2} f(x) = -1$

5. $\lim\limits_{x \to 3^-} f(x)$

This is the limit as $x$ approaches 3 from the left. As $x \to 3^-$, the value of $f(x)$ approaches 0.

$\lim\limits_{x \to 3^-} f(x) = 0$

6. $\lim\limits_{x \to 3^+} f(x)$

This is the limit as $x$ approaches 3 from the right. As $x \to 3^+$, the value of $f(x)$ approaches -1.

$\lim\limits_{x \to 3^+} f(x) = -1$

7. $\lim\limits_{x \to 3} f(x)$

Since the left-hand and right-hand limits as $x \to 3$ are different (0 from the left and -1 from the right), the two-sided limit does not exist (DNE).

$\lim\limits_{x \to 3} f(x) = \text{DNE}$

8. $\lim\limits_{x \to 4} f(x)$

As $x \to 4$, the value of $f(x)$ approaches -1 from both sides.

$\lim\limits_{x \to 4} f(x) = -1$

Would you like further clarification on any of these limits or additional explanations?

Here are five related questions:

1. What is the difference between one-sided and two-sided limits?
2. Can a limit exist at a point where the function is not defined?
3. How do we determine whether a two-sided limit exists?
4. What is the significance of discontinuities in limit calculations?
5. How do limits relate to the derivative of a function?

Tip: When analyzing limits graphically, always check both the left-hand and right-hand behavior at the point of interest.