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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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08c083ee-5dc4-4392-bfaa-a5f4117ce6af%2F53289281-7688-4703-8c9f-dedf5b21853d%2F2fyocqp.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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