### Calculus Problems on Limits and Continuity The following are exercises related to limits and continuity of a function \( f(x) \): 1. **Evaluate the following limits:** a) \( \lim_{x \to 3^-} f(x) \) b) \( \lim_{x \to 3^+} f(x) \) c) \( \lim_{x \to 3} f(x) \) d) \( f(3) \) e) Is \( f(x) \) continuous at \( x = 3 \)? #### Graph Description: The graph provided is a piecewise graph of the function \( f(x) \): - The x-axis ranges from 0 to 5. - The y-axis ranges from 0 to 4. - There is a solid black dot (closed circle) at approximately (2, 2). - There is an open circle at (3, 4) indicating that \( f(3) \) is not defined at the y-value of 4. - The graph first rises from (1, 1) to (2, 2), then dips down at (1, 3), before going up again at (3, 4). - Beyond \( x = 3 \), the graph decreases again and extends to (5, 1) keeping a linear downward pattern from (3, 4) to (5, 1). ### Questions: 1. **a) \( \lim_{x \to 3^-} f(x) \)**: This limit approaches the value of the function as \( x \) approaches 3 from the left. 2. **b) \( \lim_{x \to 3^+} f(x) \)**: This limit approaches the value of the function as \( x \) approaches 3 from the right. 3. **c) \( \lim_{x \to 3} f(x) \)**: This limit considers the value the function approaches as \( x \) nears 3 from both directions. 4. **d) \( f(3) \)**: This point evaluates the function value at \( x = 3 \). 5. **e) Continuity at \( x = 3 \)**: Determining whether \( f(x) \) is continuous at \( x = 3 \), considering the function value and the
### Calculus Problems on Limits and Continuity The following are exercises related to limits and continuity of a function \( f(x) \): 1. **Evaluate the following limits:** a) \( \lim_{x \to 3^-} f(x) \) b) \( \lim_{x \to 3^+} f(x) \) c) \( \lim_{x \to 3} f(x) \) d) \( f(3) \) e) Is \( f(x) \) continuous at \( x = 3 \)? #### Graph Description: The graph provided is a piecewise graph of the function \( f(x) \): - The x-axis ranges from 0 to 5. - The y-axis ranges from 0 to 4. - There is a solid black dot (closed circle) at approximately (2, 2). - There is an open circle at (3, 4) indicating that \( f(3) \) is not defined at the y-value of 4. - The graph first rises from (1, 1) to (2, 2), then dips down at (1, 3), before going up again at (3, 4). - Beyond \( x = 3 \), the graph decreases again and extends to (5, 1) keeping a linear downward pattern from (3, 4) to (5, 1). ### Questions: 1. **a) \( \lim_{x \to 3^-} f(x) \)**: This limit approaches the value of the function as \( x \) approaches 3 from the left. 2. **b) \( \lim_{x \to 3^+} f(x) \)**: This limit approaches the value of the function as \( x \) approaches 3 from the right. 3. **c) \( \lim_{x \to 3} f(x) \)**: This limit considers the value the function approaches as \( x \) nears 3 from both directions. 4. **d) \( f(3) \)**: This point evaluates the function value at \( x = 3 \). 5. **e) Continuity at \( x = 3 \)**: Determining whether \( f(x) \) is continuous at \( x = 3 \), considering the function value and the
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:### Calculus Problems on Limits and Continuity
The following are exercises related to limits and continuity of a function \( f(x) \):
1. **Evaluate the following limits:**
a) \( \lim_{x \to 3^-} f(x) \)
b) \( \lim_{x \to 3^+} f(x) \)
c) \( \lim_{x \to 3} f(x) \)
d) \( f(3) \)
e) Is \( f(x) \) continuous at \( x = 3 \)?
#### Graph Description:
The graph provided is a piecewise graph of the function \( f(x) \):
- The x-axis ranges from 0 to 5.
- The y-axis ranges from 0 to 4.
- There is a solid black dot (closed circle) at approximately (2, 2).
- There is an open circle at (3, 4) indicating that \( f(3) \) is not defined at the y-value of 4.
- The graph first rises from (1, 1) to (2, 2), then dips down at (1, 3), before going up again at (3, 4).
- Beyond \( x = 3 \), the graph decreases again and extends to (5, 1) keeping a linear downward pattern from (3, 4) to (5, 1).
### Questions:
1. **a) \( \lim_{x \to 3^-} f(x) \)**: This limit approaches the value of the function as \( x \) approaches 3 from the left.
2. **b) \( \lim_{x \to 3^+} f(x) \)**: This limit approaches the value of the function as \( x \) approaches 3 from the right.
3. **c) \( \lim_{x \to 3} f(x) \)**: This limit considers the value the function approaches as \( x \) nears 3 from both directions.
4. **d) \( f(3) \)**: This point evaluates the function value at \( x = 3 \).
5. **e) Continuity at \( x = 3 \)**: Determining whether \( f(x) \) is continuous at \( x = 3 \), considering the function value and the
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