**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

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

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**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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