The image you provided shows a graph representing a function $f(x)$, and you have a set of limits to evaluate based on this graph. Let's go through each one:

a) $\lim_{x \to 0^-} f(x)$

• This is the left-hand limit as $x$ approaches 0.
• We examine the behavior of $f(x)$ from the left side of 0 on the graph.
• The graph shows that as $x \to 0^-$, $f(x)$ approaches a value of 2.

b) $\lim_{x \to 0^+} f(x)$

• This is the right-hand limit as $x$ approaches 0.
• We examine the behavior of $f(x)$ from the right side of 0 on the graph.
• The graph shows that as $x \to 0^+$, $f(x)$ approaches a value of -1.

c) $\lim_{x \to 0} f(x)$

• This is the two-sided limit as $x$ approaches 0.
• Since the left-hand limit and the right-hand limit are not equal, the overall limit does not exist at $x = 0$.

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

• This is the limit as $x$ approaches 1.
• From the graph, as $x \to 1$, the value of $f(x)$ appears to be 2.

e) $\lim_{x \to 3} f(x)$

• This is the limit as $x$ approaches 3.
• From the graph, as $x \to 3$, $f(x)$ approaches 1.

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

• This is the limit as $x$ approaches -1.
• From the graph, as $x \to -1$, $f(x)$ approaches -1.

Would you like further details or clarifications on any of these steps? Here are some additional questions for exploration:

1. What does it mean when a limit does not exist at a point?
2. How can you determine one-sided limits graphically?
3. What happens to the function if the left and right limits are different?
4. How do limits at infinity differ from limits at finite points?
5. How do you handle discontinuities when evaluating limits?

Tip: When evaluating limits from a graph, always check if the left-hand and right-hand limits match. If they do not, the two-sided limit does not exist.