Let's analyze the graph and the limits presented in the question.

The function $f(x)$ is represented graphically, and the question asks us to evaluate the following limits:

1. $\lim\limits_{x \to 2^-} f(x)$ (left-hand limit as $x$ approaches 2)
2. $\lim\limits_{x \to 2^+} f(x)$ (right-hand limit as $x$ approaches 2)
3. $\lim\limits_{x \to 2} f(x)$ (two-sided limit as $x$ approaches 2)

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

From the graph, as $x$ approaches 2 from the left (i.e., $x \to 2^-$), the function value tends towards $3$. Therefore: $\lim_{x \to 2^-} f(x) = 3$

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

As $x$ approaches 2 from the right (i.e., $x \to 2^+$), the function value tends towards $1$. Therefore: $\lim_{x \to 2^+} f(x) = 1$

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

The two-sided limit exists only if the left-hand and right-hand limits are equal. In this case: $\lim_{x \to 2^-} f(x) = 3 \quad \text{and} \quad \lim_{x \to 2^+} f(x) = 1$ Since these two limits are not equal, the two-sided limit does not exist.

Final Answers:

1. $\lim_{x \to 2^-} f(x) = 3$
2. $\lim_{x \to 2^+} f(x) = 1$
3. $\lim_{x \to 2} f(x)$ does not exist.

Would you like further details on any of the steps?

Here are some related questions:

1. What is the importance of checking both left-hand and right-hand limits?
2. How can a function be continuous at a point if the two-sided limit does not exist?
3. What types of discontinuities does this graph show at $x = 2$?
4. Can a function have different limits from the left and right but still be continuous?
5. How do removable and jump discontinuities differ?

Tip: Always check both the left-hand and right-hand limits when determining if a two-sided limit exists.