https://file.aiquickdraw.com/mathbot/file/1727055640322s5y65s56.jpg

Let's analyze the graph step by step to answer the questions given.

### Analyzing the Graph
The graph represents the function \( f(x) \) with several key points where it has specific behaviors that we will evaluate.

#### **a) Calculating the left and right limits at the given points:**

1. **\(\lim_{x \to 0^-} f(x)\)**: 
   Observing the graph as \( x \) approaches 0 from the left, \( f(x) \) approaches 2.

2. **\(\lim_{x \to 0^+} f(x)\)**:
   As \( x \to 0^+ \), \( f(x) \) still approaches 2. Therefore,
   \[
   \lim_{x \to 0} f(x) = 2.
   \]

3. **\(\lim_{x \to 18^-} f(x)\)**:
   As \( x \to 18 \) from the left, \( f(x) \) approaches the value \( 6 \).

4. **\(\lim_{x \to 18^+} f(x)\)**:
   From the right side of \( x = 18 \), the function approaches the value \( 4 \).

   Therefore, since the left-hand and right-hand limits do not match at \( x = 18 \), the overall limit does not exist at this point.

### Summary of limits:
- \( \lim_{x \to 0} f(x) = 2 \)
- \( \lim_{x \to 18^-} f(x) = 6 \)
- \( \lim_{x \to 18^+} f(x) = 4 \)

#### **b) At which of the points \( x = 0, 16, 18, 19 \) does the limit of \( f(x) \) exist?**

From the observations:
- At \( x = 0 \), the left-hand and right-hand limits are equal: **exists**.
- At \( x = 16 \), the function does not show any jump or discontinuity, so the limit exists.
- At \( x = 18 \), the left and right-hand limits differ: **does not exist**.
- At \( x = 19 \), the function appears to have a definite value from both sides.

Thus, the limit exists at: **\( x = 0, 16, 19 \)**.

#### **c) At which of the points \( x = 0, 16, 18, 19 \) is the function continuous?**

A function is continuous at a point \( x = a \) if:
1. The limit \( \lim_{x \to a} f(x) \) exists.
2. \( f(a) \) exists.
3. The limit equals \( f(a) \).

- At \( x = 0 \), \( f(0) = 2 \), and the limit \( \lim_{x \to 0} f(x) = 2 \). Hence, \( f(x) \) is continuous at \( x = 0 \).
- At \( x = 16 \), there is no jump or break, and \( f(x) \) appears to be continuous.
- At \( x = 18 \), the function has a jump discontinuity since left and right-hand limits differ. So it is **not continuous**.
- At \( x = 19 \), although there is a closed dot, the function is continuous as both left-hand and right-hand limits match the function value at \( x = 19 \).

Therefore, the function is continuous at: **\( x = 0, 16, 19 \)**.

### Summary of Answers

**a)**
- \( \lim_{x \to 0} f(x) = 2 \)
- \( \lim_{x \to 18^-} f(x) = 6 \)
- \( \lim_{x \to 18^+} f(x) = 4 \) 
  (Thus, diverges overall at \( x = 18 \))

**b)** The limit exists at \( x = 0; 16; 19 \).

**c)** The function is continuous at \( x = 0; 16; 19 \).

Would you like more details or have any further questions?

### **5 Relative Questions:**
1. How do left-hand and right-hand limits impact overall limit existence?
2. What conditions make a function discontinuous at a certain point?
3. What other types of discontinuities can occur besides jumps?
4. How can graph analysis help determine function behavior at specific points?
5. How does continuity at a point relate to differentiability?

### **Tip:**
Always check left-hand and right-hand limits separately to confirm whether a function is continuous or the overall limit exists.

Consider the graph of the function f below. All interesting features have integer coordinates. a) Calculate the left and right limits of these functions at the various points specified. Write diverges if the limit does not exist. b) At which of the points x = 0; 16; 18; 19 does the limit of f exist? c) At which of the points x = 0; 16; 18; 19 is the function continuous?

Analyze the function f to determine its left-hand and right-hand limits at points x = 0, 16, 18, and 19. Explore concepts of continuity and limits using the provided graph.

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