Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### 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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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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