### Estimating Limits Using Graphing Utilities#### Instructions:1. **Objective:** Use a graphing utility to complete the table and estimate the limit as $ x $ approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates. (Round your answers to one decimal place.)2. **Function Provided:** \[ f(x) = x^2 - x \sqrt{x^2 - x} \]3. **Table for Estimation:** Complete the table using the graphing utility: | $ x $ | $ 10^0 $ | $ 10^1 $ | $ 10^2 $ | $ 10^3 $ | |:--------:|:----------:|:----------:|:----------:|:----------:| | $ f(x) $ | | | | | | $ x $ | $ 10^4 $ | $ 10^5 $ | $ 10^6 $ | |:--------:|:----------:|:----------:|:----------:| | $ f(x) $ | | | |4. **Limit Calculation:** Find the limit analytically and fill in the value below: \[ \lim_{x \to \infty} \quad = \quad \boxed{} \]### Explanation for Graph or Diagram:Since no graphical diagram is provided within the problem itself, the explanation assumes you will use a graphing utility to visualize the function $ f(x) $. Here's a detailed step-by-step guide:1. **Graphing Steps:** - Input the function $ f(x) = x^2 - x \sqrt{x^2 - x} $ into your graphing utility. - Set the x-axis to display a range from $ 10^0 $ to $ 10^6 $. This helps to visualize the behavior as $ x $ approaches infinity. 2. **Observation:** - As you analyze the graph, observe how the function behaves as $ x $ increases. - Use the graphing utility to estimate the value the function approaches (the limit) as $ x $ becomes very large.

Answered: Image /qna-images/answer/53289281-7688-4703-8c9f-dedf5b21853d.jpg

Answered: s(x) = - V – 1 100 10! 102 103 F (x)…

Math

Calculus

s(x) = - V – 1 100 10! 102 103 F (x) 104 105 106 f (x) lim

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

### Estimating Limits Using Graphing Utilities

#### Instructions:
1. **Objective:**
Use a graphing utility to complete the table and estimate the limit as $ x $ approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates. (Round your answers to one decimal place.)

2. **Function Provided:**
\[
f(x) = x^2 - x \sqrt{x^2 - x}
\]

3. **Table for Estimation:**
Complete the table using the graphing utility:

| $ x $ | $ 10^0 $ | $ 10^1 $ | $ 10^2 $ | $ 10^3 $ |
|:--------:|:----------:|:----------:|:----------:|:----------:|
| $ f(x) $ | | | | |

| $ x $ | $ 10^4 $ | $ 10^5 $ | $ 10^6 $ |
|:--------:|:----------:|:----------:|:----------:|
| $ f(x) $ | | | |

4. **Limit Calculation:**
Find the limit analytically and fill in the value below:
\[
\lim_{x \to \infty} \quad = \quad \boxed{}
\]

### Explanation for Graph or Diagram:
Since no graphical diagram is provided within the problem itself, the explanation assumes you will use a graphing utility to visualize the function $ f(x) $. Here's a detailed step-by-step guide:

1. **Graphing Steps:**
- Input the function $ f(x) = x^2 - x \sqrt{x^2 - x} $ into your graphing utility.
- Set the x-axis to display a range from $ 10^0 $ to $ 10^6 $. This helps to visualize the behavior as $ x $ approaches infinity.

2. **Observation:**
- As you analyze the graph, observe how the function behaves as $ x $ increases.
- Use the graphing utility to estimate the value the function approaches (the limit) as $ x $ becomes very large.

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