To solve these problems, we need to analyze the behavior of the function $f(x)$ around the point $x = 1$ by looking at the graph you provided.

a) $\lim_{x \to 1} f(x)$

This is the limit of $f(x)$ as $x$ approaches 1 from both sides (left and right). If the left-hand limit (as $x \to 1^-$) and the right-hand limit (as $x \to 1^+$) are equal, then the two-sided limit exists and is equal to this value.

b) $\lim_{x \to 1^-} f(x)$

This is the limit of $f(x)$ as $x$ approaches 1 from the left (values of $x$ slightly less than 1). On the graph, we look at how $f(x)$ behaves as $x$ gets closer to 1 from the negative side.

c) $\lim_{x \to 1^+} f(x)$

This is the limit of $f(x)$ as $x$ approaches 1 from the right (values of $x$ slightly greater than 1). We analyze the graph to see the behavior of $f(x)$ as $x$ gets closer to 1 from the positive side.

Analysis of the Graph:

From the graph:

1. Left-hand limit (as $x \to 1^-$): As $x$ approaches 1 from the left, the function $f(x)$ is approaching the value $y = -2$. So: $\lim_{x \to 1^-} f(x) = -2$

2. Right-hand limit (as $x \to 1^+$): As $x$ approaches 1 from the right, the function $f(x)$ is approaching the value $y = 4$. So: $\lim_{x \to 1^+} f(x) = 4$

3. Two-sided limit (as $x \to 1$): Since the left-hand limit and right-hand limit are not equal, the two-sided limit does not exist: $\lim_{x \to 1} f(x) \text{ does not exist.}$

Conclusion:

• a) $\lim_{x \to 1} f(x)$: Does not exist.
• b) $\lim_{x \to 1^-} f(x) = -2$
• c) $\lim_{x \to 1^+} f(x) = 4$

Would you like to explore more details about these limits or have any specific questions? Here are a few related questions you might consider:

1. What is the significance of left-hand and right-hand limits in determining continuity?
2. How do we identify discontinuities in a function from its graph?
3. Can you describe how to find the limit of a function as $x \to a$ when there are jumps in the graph?
4. How would the graph look if $\lim_{x \to 1} f(x)$ existed?
5. What are different types of discontinuities and how do they affect limits?

Tip: When evaluating limits from a graph, always check for the behavior from both sides of the point in question.