https://file.aiquickdraw.com/mathbot/file/1726203074134562aaul2.jpg

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?

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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.

The graph of the function f is shown in the figure. Which of the following statements about f is true?

Understand how to evaluate limits and continuity from a graph. This problem explains key concepts such as left-hand and right-hand limits, discontinuities, and limit existence using a graphical approach. Suitable for students in Grades 10-12.

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