Let's analyze the graph and the question to determine which of the given statements is true.

Step 1: Examine the graph

In the graph, we see two key points labeled $a$ and $b$ on the x-axis:

• At $x = a$, there is a discontinuity. The graph approaches different values from the left and right.
• At $x = b$, the function seems continuous and has a defined value.

Step 2: Analyze limits based on the graph

For $x = a$:

• The left-hand limit as $x \to a^-$ seems to approach 3.
• The right-hand limit as $x \to a^+$ seems to approach 1.
• Since the left-hand limit and right-hand limit are not equal, the limit as $x \to a$ does not exist.

For $x = b$:

• The graph is continuous at $x = b$.
• The value of the function $f(b)$ appears to be $-1$.
• The limit as $x \to b$ is $-1$, which is also the value of the function at $x = b$.

Step 3: Evaluate the statements

Now we can check the statements:

A) $\lim_{x \to a} f(x) = \lim_{x \to b} f(x)$: This is false because the limit as $x \to a$ does not exist.

B) $\lim_{x \to a} f(x) = 3$: This is false because the limit as $x \to a$ does not exist (the left-hand and right-hand limits are different).

C) $\lim_{x \to b} f(x) = 3$: This is false because the limit as $x \to b$ is $-1$, not 3.

D) $\lim_{x \to b} f(x) = -1$: This is true since the limit as $x \to b$ is $-1$.

E) $\lim_{x \to a} f(x)$ does not exist: This is true since the left-hand and right-hand limits as $x \to a$ are different.

Step 4: Conclusion

The correct answers are D and E.

Would you like further details on any part of the solution?

Here are some follow-up questions for you:

1. What are the key conditions for the existence of a limit at a point?
2. How would the graph change if the limit at $x = a$ existed?
3. Can a function be discontinuous but still have a limit at a point?
4. How does the graph of a function help in understanding one-sided limits?
5. What are the implications of having equal left-hand and right-hand limits?

Tip: When evaluating limits from a graph, always check for one-sided limits to see if the limit exists at a given point.