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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![### Understanding Limits
The study of limits is fundamental in calculus and mathematical analysis, involving the behavior of a function as its argument approaches a particular value.
---
#### Example Problem:
**Find the following limit:**
\[ \lim_{{x \to +\infty}} f(x) \]
Where \( f(x) \) is described by the graph provided.
#### Graph Explanation
The graph shows a curve for the function \( f(x) \) across different values of \( x \). The x-axis and y-axis are marked in intervals.
To understand the limit of \( f(x) \) as \( x \) approaches positive infinity:
1. **Observe the right-hand side of the graph**:
- As \( x \) increases (moves to the right), analyze the behavior of \( f(x) \).
2. **Visualization**:
- Initially, \( f(x) \) exhibits fluctuations, but for large values of \( x \), \( f(x) \) trends drastically upwards.
Given the rapid upward trend in the function as \( x \) heads towards infinity, the limit approaches infinity.
#### Answer Choices:
A. \( \infty \)
B. 0
C. 3
D. 4
**Correct Answer: A. \( \infty \)**
Understanding this concept helps in analyzing the long-term behavior of functions and is crucial in calculus.
---
**Test Yourself**:
Now try finding the limit for other values approaching infinity with different functions to solidify your understanding!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd9966fd-53bb-4986-9621-0cf929ae8043%2Fd40a4fff-075e-4fda-98ff-68dc999f8641%2Fr2gxjqt_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Limits
The study of limits is fundamental in calculus and mathematical analysis, involving the behavior of a function as its argument approaches a particular value.
---
#### Example Problem:
**Find the following limit:**
\[ \lim_{{x \to +\infty}} f(x) \]
Where \( f(x) \) is described by the graph provided.
#### Graph Explanation
The graph shows a curve for the function \( f(x) \) across different values of \( x \). The x-axis and y-axis are marked in intervals.
To understand the limit of \( f(x) \) as \( x \) approaches positive infinity:
1. **Observe the right-hand side of the graph**:
- As \( x \) increases (moves to the right), analyze the behavior of \( f(x) \).
2. **Visualization**:
- Initially, \( f(x) \) exhibits fluctuations, but for large values of \( x \), \( f(x) \) trends drastically upwards.
Given the rapid upward trend in the function as \( x \) heads towards infinity, the limit approaches infinity.
#### Answer Choices:
A. \( \infty \)
B. 0
C. 3
D. 4
**Correct Answer: A. \( \infty \)**
Understanding this concept helps in analyzing the long-term behavior of functions and is crucial in calculus.
---
**Test Yourself**:
Now try finding the limit for other values approaching infinity with different functions to solidify your understanding!
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