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: Evaluate each of the following limits.
![### Limits at Infinity: Example Problems
This section provides a closer look at limits involving infinity, with two example problems to illustrate key concepts.
---
#### Example Problems
---
**(a)**
\[ \lim_{{x \to \infty}} \left( x - \sqrt{x^2 + 7} \right) \]
**(b)**
\[ \lim_{{x \to -\infty}} \left( x - \sqrt{x^2 + 7} \right) \]
---
### Explanation of Problems:
These example problems involve finding the limits of expressions as \( x \) approaches positive infinity and negative infinity. The expressions feature a variable \( x \) subtracted from the square root of a quadratic polynomial.
#### Analysis:
1. **Behavior at Positive Infinity:**
- In example (a), as \( x \) approaches \(\infty\), we analyze the behavior of the expression \( x - \sqrt{x^2 + 7} \).
2. **Behavior at Negative Infinity:**
- In example (b), as \( x \) approaches \(-\infty\), we perform a similar analysis for the expression \( x - \sqrt{x^2 + 7} \).
### Objective:
Understanding these limit calculations helps illustrate the behavior of expressions involving roots and quadratic polynomials as the variable \( x \) moves towards extremely large positive and negative values.
For detailed solutions and step-by-step explanations, continue reading or consult the related instructional videos provided in this section.
---
This transcription is tailored for educational purposes, aiming to support students in grasping the fundamental concepts of limits at infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc965ec1f-6ecd-467d-b5f1-022eb1dfa0a4%2F1eaf886c-60ea-4f0a-ae25-9d6d4d20521c%2Fvk2o8k9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Limits at Infinity: Example Problems
This section provides a closer look at limits involving infinity, with two example problems to illustrate key concepts.
---
#### Example Problems
---
**(a)**
\[ \lim_{{x \to \infty}} \left( x - \sqrt{x^2 + 7} \right) \]
**(b)**
\[ \lim_{{x \to -\infty}} \left( x - \sqrt{x^2 + 7} \right) \]
---
### Explanation of Problems:
These example problems involve finding the limits of expressions as \( x \) approaches positive infinity and negative infinity. The expressions feature a variable \( x \) subtracted from the square root of a quadratic polynomial.
#### Analysis:
1. **Behavior at Positive Infinity:**
- In example (a), as \( x \) approaches \(\infty\), we analyze the behavior of the expression \( x - \sqrt{x^2 + 7} \).
2. **Behavior at Negative Infinity:**
- In example (b), as \( x \) approaches \(-\infty\), we perform a similar analysis for the expression \( x - \sqrt{x^2 + 7} \).
### Objective:
Understanding these limit calculations helps illustrate the behavior of expressions involving roots and quadratic polynomials as the variable \( x \) moves towards extremely large positive and negative values.
For detailed solutions and step-by-step explanations, continue reading or consult the related instructional videos provided in this section.
---
This transcription is tailored for educational purposes, aiming to support students in grasping the fundamental concepts of limits at infinity.
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