https://file.aiquickdraw.com/mathbot/file/1730994320270pfiwyjt6.jpg

Let's solve the problem based on the information provided:

The problem involves evaluating certain values and limits from the graph of \( f(x) \) shown in Figure 13.

1. **\( f(-3) \)**:
   - From the graph, at \( x = -3 \), there is a filled dot at \( y = -2 \).
   - Therefore, \( f(-3) = -2 \).

2. **\( f(3) \)**:
   - At \( x = 3 \), there is an open circle at \( y = 1 \), meaning \( f(3) \) does not exist since there’s no defined value at that point.
   - So, \( f(3) \) does not exist.

3. **\( \lim_{x \to -3^+} f(x) \)**:
   - As \( x \) approaches \( -3 \) from the right, the function value approaches \( y = 1 \).
   - Thus, \( \lim_{x \to -3^+} f(x) = 1 \).

4. **\( \lim_{x \to -3^-} f(x) \)**:
   - As \( x \) approaches \( -3 \) from the left, the function value approaches \( y = -2 \).
   - Therefore, \( \lim_{x \to -3^-} f(x) = -2 \).

5. **\( \lim_{x \to -3} f(x) \)**:
   - Since \( \lim_{x \to -3^+} f(x) \neq \lim_{x \to -3^-} f(x) \), the two-sided limit \( \lim_{x \to -3} f(x) \) does not exist.

6. **\( \lim_{x \to 3} f(x) \)**:
   - As \( x \) approaches \( 3 \) from both the left and the right, \( f(x) \) approaches \( y = 1 \).
   - Thus, \( \lim_{x \to 3} f(x) = 1 \).

### Summary of Results:
- (a) \( f(-3) = -2 \)
- (b) \( f(3) \) does not exist
- (c) \( \lim_{x \to -3^+} f(x) = 1 \)
- (d) \( \lim_{x \to -3^-} f(x) = -2 \)
- (e) \( \lim_{x \to -3} f(x) \) does not exist
- (f) \( \lim_{x \to 3} f(x) = 1 \)

Would you like a more detailed explanation of any of these answers?

Here are five related questions to deepen your understanding:

1. How does the existence of open or closed circles at certain points affect the limits and function values?
2. Why does the limit at \( x = -3 \) not exist if the left- and right-hand limits differ?
3. How do you determine one-sided limits by looking at a graph?
4. What is the significance of a function value not existing at a certain point, as in \( f(3) \)?
5. How would the answer change if there were a closed dot at \( x = 3 \) on the graph?

**Tip:** When evaluating limits graphically, always check both the left-hand and right-hand behavior as \( x \) approaches the point of interest.

For the function f graphed in Figure 13, find the indicated limit or function value, or state that it does not exist:
(a) f(-3)
(b) f(3)
(c) lim x→-3+ f(x)
(d) lim x→-3- f(x)
(e) lim x→-3 f(x)
(f) lim x→3 f(x).

This problem involves finding limits and function values from a graph. Explore key limit concepts, including one-sided limits and the conditions for two-sided limit existence, with a step-by-step breakdown for f(x) graphed in Figure 13. Suitable for high school calculus 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