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Give a formula for a function f(x) that has a limit as x approaches 3 but for which the function is not defined at x=3. Graph this function and explain your reasoning. Give a formula for a function that has a limit as x approaches 3 but the value of the function f(3) is not equal to the limit. Explain your thinking. Give a formula for a function that has no limit as x approaches 3. Explain.

Give a formula for a function f(x) that has a limit as x approaches 3 but for which the function is not defined at x=3. Graph this function and explain your reasoning. Give a formula for a function that has a limit as x approaches 3 but the value of the function f(3) is not equal to the limit. Explain your thinking. Give a formula for a function that has no limit as x approaches 3. Explain.

Calculus: Early Transcendentals
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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**Text Transcription for Educational Website:** --- 1. **Problem:** Give a formula for a function \( f(x) \) that has a limit as \( x \) approaches 3 but for which the function is not defined at \( x = 3 \). Graph this function and explain your reasoning. 2. **Problem:** Give a formula for a function that has a limit as \( x \) approaches 3 but the value of the function \( f(3) \) is not equal to the limit. Explain your thinking. 3. **Problem:** Give a formula for a function that has no limit as \( x \) approaches 3. Explain. --- ### Explanation Guide: - **Problem 1:** Consider \( f(x) = \frac{x^2 - 9}{x - 3} = \frac{(x+3)(x-3)}{x-3} \). The function is not defined at \( x = 3 \) because it results in division by zero. However, the limit as \( x \) approaches 3 is equal to \( 6 \), which can be determined by simplification and ignoring the hole at \( x = 3 \). - **Problem 2:** For instance, \( f(x) = \begin{cases} x^2 - 1 & \text{if } x \neq 3 \\ 10 & \text{if } x = 3 \end{cases} \). The limit as \( x \) approaches 3 is \( 8 \) (since \( 9 - 1 = 8 \)), but \( f(3) = 10 \). This example represents a point discontinuity. - **Problem 3:** An example of this can be \( f(x) = \frac{1}{(x-3)^2} \). As \( x \) approaches 3, the function becomes unbounded, meaning the limit does not exist due to the vertical asymptote at \( x = 3 \). ### Graphical Representation: When creating graphs for each scenario: - **Problem 1 Graph:** Show a straight line \( y = x + 3 \) with a hole at \( x = 3 \). - **Problem 2 Graph:** Display \( y = x^2 - 1 \) and mark a distinct point at \( (3, 10) \) to indicate the
Transcribed Image Text:**Text Transcription for Educational Website:** --- 1. **Problem:** Give a formula for a function \( f(x) \) that has a limit as \( x \) approaches 3 but for which the function is not defined at \( x = 3 \). Graph this function and explain your reasoning. 2. **Problem:** Give a formula for a function that has a limit as \( x \) approaches 3 but the value of the function \( f(3) \) is not equal to the limit. Explain your thinking. 3. **Problem:** Give a formula for a function that has no limit as \( x \) approaches 3. Explain. --- ### Explanation Guide: - **Problem 1:** Consider \( f(x) = \frac{x^2 - 9}{x - 3} = \frac{(x+3)(x-3)}{x-3} \). The function is not defined at \( x = 3 \) because it results in division by zero. However, the limit as \( x \) approaches 3 is equal to \( 6 \), which can be determined by simplification and ignoring the hole at \( x = 3 \). - **Problem 2:** For instance, \( f(x) = \begin{cases} x^2 - 1 & \text{if } x \neq 3 \\ 10 & \text{if } x = 3 \end{cases} \). The limit as \( x \) approaches 3 is \( 8 \) (since \( 9 - 1 = 8 \)), but \( f(3) = 10 \). This example represents a point discontinuity. - **Problem 3:** An example of this can be \( f(x) = \frac{1}{(x-3)^2} \). As \( x \) approaches 3, the function becomes unbounded, meaning the limit does not exist due to the vertical asymptote at \( x = 3 \). ### Graphical Representation: When creating graphs for each scenario: - **Problem 1 Graph:** Show a straight line \( y = x + 3 \) with a hole at \( x = 3 \). - **Problem 2 Graph:** Display \( y = x^2 - 1 \) and mark a distinct point at \( (3, 10) \) to indicate the
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