https://file.aiquickdraw.com/mathbot/file/1726162610087c6znzebl.jpg

Answer the questions correctly and fill them into the blank spaces in the response segment.

To determine whether the function \( f(x) \) is continuous at \( x = -3 \), we will follow the definition of continuity, which requires three conditions to be met:

1. **The limit from the left** and **the limit from the right** at \( x = -3 \) must exist and be equal (i.e., \( \lim_{x \to -3^{-}} f(x) = \lim_{x \to -3^{+}} f(x) \)).
2. The function value \( f(-3) \) must exist.
3. The value of the limit \( \lim_{x \to -3} f(x) \) must equal \( f(-3) \).

### Step 1: Evaluate the Left-Hand and Right-Hand Limits
From the graph:
- As \( x \to -3^{-} \), the graph approaches \( y = 2 \). Hence, \( \lim_{x \to -3^{-}} f(x) = 2 \).
- As \( x \to -3^{+} \), the graph also approaches \( y = 2 \). Hence, \( \lim_{x \to -3^{+}} f(x) = 2 \).

Since both the left-hand and right-hand limits are equal, we can conclude:
\[
\lim_{x \to -3} f(x) = 2.
\]

### Step 2: Evaluate \( f(-3) \)
Looking at the graph at \( x = -3 \), we see that there is a closed dot at \( ( -3, 2 ) \). Therefore, \( f(-3) = 2 \).

### Step 3: Check if the Function is Continuous at \( x = -3 \)
Since the limit as \( x \to -3 \) exists and is equal to \( f(-3) \), the function is continuous at \( x = -3 \).

### Fill in the Blanks:
- \( \lim_{x \to -3^{-}} f(x) = 2 \)
- \( \lim_{x \to -3^{+}} f(x) = 2 \)
- \( \lim_{x \to -3} f(x) = 2 \)
- \( f(-3) = 2 \)
- Therefore, the function \( f(x) \) **is continuous** at \( x = -3 \) because **the limit as \( x \) approaches -3 is equal to \( f(-3) \)**.

Let me know if you want more details or explanations!

---

Here are some related questions to consider:
1. What happens if \( f(-3) \) does not equal the limit as \( x \to -3 \)?
2. How do you determine the continuity of a piecewise function at the boundary points?
3. What are the three conditions for continuity at a point?
4. How do left-hand and right-hand limits differ conceptually?
5. Can a function be continuous everywhere but not differentiable?

**Tip:** Always check for holes or breaks in the graph when determining limits!

Use the definition of continuity to determine whether the function f(x) graphed below is continuous at x = -3.

Learn how to use the definition of continuity to determine if a function is continuous at x = -3. This problem covers the evaluation of left-hand and right-hand limits and checking if the function value equals the limit. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation