enter each answer as a whole number (like -4, 0, or 253) or dne for undefined or does not exist.
$f(5) = \square$
$\lim\limits_{x \to 3^-} f(x) = \square$
$\lim\limits_{x \to 9^+} f(x) = \square$
$\lim\limits_{x \to 7} f(x) = \square$

Question

Accepted Answer

### 1. Solve $ f(5) $
# Explanation:
## Step1: Find the point at $ x = 5 $
To find $ f(5) $, we look at the graph of $ y = f(x) $ and find the $ y $-value corresponding to $ x = 5 $. From the graph, at $ x = 5 $, the filled dot (representing the function's value) is at $ y = 1 $? Wait, no, wait. Wait, the open circle and closed circle: at $ x = 5 $, the closed dot (the solid point) is at $ y = 1 $? Wait, no, looking again: at $ x = 5 $, there's a closed dot (solid) at $ y = 1 $ and an open dot (hollow) at $ y = 2 $. Wait, no, the function's value $ f(5) $ is the value of the solid dot (the defined point) at $ x = 5 $. Wait, no, actually, in piecewise functions, the solid dot is the one that defines the function's value. Wait, looking at the graph: at $ x = 5 $, the solid (filled) circle is at $ y = 1 $, and the open circle is at $ y = 2 $. Wait, no, maybe I misread. Wait, the graph: at $ x = 5 $, there's a filled dot (red solid) at $ y = 1 $ and an open dot (red hollow) at $ y = 2 $. So $ f(5) $ is the value of the solid dot, which is $ 1 $? Wait, no, wait, maybe I got it wrong. Wait, let's check again. Wait, the open circles are the limits, and the closed circles are the function's value. Wait, no, actually, in a graph, the closed dot (filled circle) represents the point that is included in the function, so $ f(x) $ at $ x = a $ is the $ y $-value of the closed dot at $ x = a $. Wait, at $ x = 5 $, the closed dot (solid) is at $ y = 1 $, and the open dot (hollow) is at $ y = 2 $. So $ f(5) = 1 $? Wait, no, maybe I made a mistake. Wait, let's look at the coordinates: at $ x = 5 $, the solid point (filled) is at $ (5, 1) $, and the open point (hollow) is at $ (5, 2) $. So $ f(5) = 1 $.

## Step2: Confirm the value
So from the graph, when $ x = 5 $, the function's value (the filled dot) is $ 1 $. So $ f(5) = 1 $.

# Answer: $ 1 $

### 2. Solve $ \lim_{x \to 3^-} f(x) $
# Explanation:
## Step1: Understand left-hand limit
The left-hand limit as $ x \to 3^- $ means we approach $ x = 3 $ from the left (values less than 3). So we look at the graph as $ x $ approaches 3 from the left side.

## Step2: Analyze the graph from the left of $ x = 3 $
From the left of $ x = 3 $ (i.e., as $ x $ gets closer to 3 from values like 2, 2.5, etc.), the graph is a line going from $ x = 1 $ (open dot at $ (1, 1) $) to $ x = 3 $ (open dot at $ (3, 3) $)? Wait, no, at $ x = 1 $, there's an open dot at $ (1, 1) $ and a line going up to $ x = 3 $, open dot at $ (3, 3) $. Wait, when approaching $ x = 3 $ from the left, the $ y $-value approaches 3? Wait, no, let's see: from $ x = 1 $ (open dot at $ (1, 1) $) to $ x = 3 $ (open dot at $ (3, 3) $), the line is increasing. So as $ x $ approaches 3 from the left (values less than 3, moving towards 3), the $ y $-value approaches the open dot at $ x = 3 $, which is 3. Wait, the open dot at $ x = 3 $ (left side) is at $ y = 3 $? Wait, at $ x = 3 $, there's an open dot (hollow) at $ y = 3 $ and a closed dot (filled) at $ y = 2 $. Wait, no, when approaching $ x = 3 $ from the left, we follow the line that comes from the left. The line from $ x = 1 $ (open dot $ (1, 1) $) to $ x = 3 $ (open dot $ (3, 3) $): so as $ x \to 3^- $, the limit is the $ y $-value of the open dot at $ x = 3 $ from the left, which is 3.

## Step3: Confirm the left-hand limit
So $ \lim_{x \to 3^-} f(x) = 3 $.

# Answer: $ 3 $

### 3. Solve $ \lim_{x \to 9^+} f(x) $
# Explanation:
## Step1: Understand right-hand limit
The right-hand limit as $ x \to 9^+ $ means we approach $ x = 9 $ from the right (values greater than 9). From the graph, at $ x = 9 $, there's a closed dot (filled circle) at $ y = 3 $, and the graph continues to the right as a horizontal line. So as $ x $ approaches 9 from the right (values like 9.1, 9.2, etc.), the $ y $-value remains 3, because the graph is horizontal at $ y = 3 $ for $ x \geq 9 $.

## Step2: Confirm the right-hand limit
Thus, $ \lim_{x \to 9^+} f(x) = 3 $.

# Answer: $ 3 $

### 4. Solve $ \lim_{x \to 7} f(x) $
# Explanation:
## Step1: Understand the limit as $ x \to 7 $
To find $ \lim_{x \to 7} f(x) $, we need to check the left-hand limit (as $ x \to 7^- $) and the right-hand limit (as $ x \to 7^+ $) and see if they are equal.

## Step2: Find the left-hand limit as $ x \to 7^- $
As $ x $ approaches 7 from the left (values less than 7, moving towards 7), we look at the graph. The line approaching $ x = 7 $ from the left is the line going up to $ x = 7 $ (open dot? Wait, no, at $ x = 7 $, there's an open dot? Wait, no, at $ x = 7 $, the graph has a closed dot (filled circle) at $ y = 6 $. Wait, the graph: from $ x = 6 $ to $ x = 7 $, the line goes up to $ x = 7 $, where there's a closed dot at $ y = 6 $. Then from $ x = 7 $ to $ x = 9 $, the line goes down to $ x = 9 $ (open dot at $ y = 4 $). Wait, no, let's see: as $ x \to 7^- $, the $ y $-value approaches the closed dot at $ x = 7 $, which is 6? Wait, no, the left-hand limit is the value as $ x $ approaches 7 from the left, and the right-hand limit is as $ x $ approaches 7 from the right. Wait, actually, at $ x = 7 $, the graph has a peak at $ y = 6 $ (closed dot). So as $ x $ approaches 7 from the left (from values less than 7, like 6.9, 6.99, etc.), the $ y $-value approaches 6. As $ x $ approaches 7 from the right (from values greater than 7, like 7.1, 7.01, etc.), the $ y $-value also approaches 6? Wait, no, from the right, the line comes from $ x = 7 $ going down to $ x = 9 $. Wait, at $ x = 7 $, the closed dot is at $ y = 6 $, and the line to the right of $ x = 7 $ starts at $ x = 7 $ (closed dot) and goes down. So as $ x \to 7^- $, the limit is 6, and as $ x \to 7^+ $, the limit is also 6 (since the line from the right starts at $ x = 7 $ with $ y = 6 $). Therefore, the two-sided limit $ \lim_{x \to 7} f(x) = 6 $.

## Step3: Confirm the limit
Since both the left-hand limit and right-hand limit as $ x \to 7 $ are equal to 6, the limit exists and is 6.

# Answer: $ 6 $

### Final Answers:
- $ f(5) = \boxed{1} $
- $ \lim_{x \to 3^-} f(x) = \boxed{3} $
- $ \lim_{x \to 9^+} f(x) = \boxed{3} $
- $ \lim_{x \to 7} f(x) = \boxed{6} $

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QUESTION IMAGE

Question

Explanation:

1. Solve \( f(5) \)

Step1: Find the point at \( x = 5 \)

Step2: Confirm the value

Step1: Understand left-hand limit

Step2: Analyze the graph from the left of \( x = 3 \)

Step3: Confirm the left-hand limit

Step1: Understand right-hand limit

Step2: Confirm the right-hand limit

Answer:

2. Solve \( \lim_{x \to 3^-} f(x) \)