Please show EVERY step.Question: Evaluate each of the following limits. ### Limits at Infinity: Example ProblemsThis 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.

Answered: Image /qna-images/answer/1eaf886c-60ea-4f0a-ae25-9d6d4d20521c.jpg

Answered: (a) lim -√²+7. FIX (b) lim *-√² +7.…

Math

Calculus

(a) lim -√²+7. FIX (b) lim *-√² +7. BITK

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

Please show EVERY step. 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.

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