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...
Related questions
Question
100%
![### Understanding Limits in Calculus
If \(\lim_{x \to 0} \frac{f(x)}{x^2} = 1\), find each of the following limits.
1. (a) \(\lim_{x \to 0} f(x)\)
- A box is provided for the students to input their answer.
- Followed by a red 'X' indicating a wrong answer or a placeholder for feedback.
2. (b) \(\lim_{x \to 0} \frac{f(x)}{x}\)
- Similarly, a box is provided for the students to input their answer.
- Followed by a red 'X' indicating a wrong answer or a placeholder for feedback.
### Explanation of the Problem
The student is asked to find the values of limits involving the function \( f(x) \) given that the limit of \( \frac{f(x)}{x^2} \) as \( x \) approaches 0 is equal to 1.
#### Understanding the Provided Information
Given:
\[ \lim_{x \to 0} \frac{f(x)}{x^2} = 1 \]
This means that as \( x \) approaches 0, \( \frac{f(x)}{x^2} \) approaches 1. Consequently, \( f(x) \) behaves similarly to \( x^2 \) multiplied by a value that gets closer to 1 as \( x \) approaches 0.
#### Detailed Steps
- To find \( \lim_{x \to 0} f(x) \):
- Since \( f(x) \) is analogous to \( x^2 \cdot 1 \) as \( x \to 0 \):
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 \cdot 1 = 0 \]
- To find \( \lim_{x \to 0} \frac{f(x)}{x} \):
- Use the given limit relation:
\[ \frac{f(x)}{x} = x \cdot \left(\frac{f(x)}{x^2}\right) \]
\[ \lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81e8ba16-e7f4-4cb8-bc35-ac7ab5472c5c%2Fac7533b3-c9a6-4a01-aa31-5cdd955c92cb%2Fah613wk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Limits in Calculus
If \(\lim_{x \to 0} \frac{f(x)}{x^2} = 1\), find each of the following limits.
1. (a) \(\lim_{x \to 0} f(x)\)
- A box is provided for the students to input their answer.
- Followed by a red 'X' indicating a wrong answer or a placeholder for feedback.
2. (b) \(\lim_{x \to 0} \frac{f(x)}{x}\)
- Similarly, a box is provided for the students to input their answer.
- Followed by a red 'X' indicating a wrong answer or a placeholder for feedback.
### Explanation of the Problem
The student is asked to find the values of limits involving the function \( f(x) \) given that the limit of \( \frac{f(x)}{x^2} \) as \( x \) approaches 0 is equal to 1.
#### Understanding the Provided Information
Given:
\[ \lim_{x \to 0} \frac{f(x)}{x^2} = 1 \]
This means that as \( x \) approaches 0, \( \frac{f(x)}{x^2} \) approaches 1. Consequently, \( f(x) \) behaves similarly to \( x^2 \) multiplied by a value that gets closer to 1 as \( x \) approaches 0.
#### Detailed Steps
- To find \( \lim_{x \to 0} f(x) \):
- Since \( f(x) \) is analogous to \( x^2 \cdot 1 \) as \( x \to 0 \):
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 \cdot 1 = 0 \]
- To find \( \lim_{x \to 0} \frac{f(x)}{x} \):
- Use the given limit relation:
\[ \frac{f(x)}{x} = x \cdot \left(\frac{f(x)}{x^2}\right) \]
\[ \lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning