For the following exercise, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. X ƒ(x) = ²x² = x 2
For the following exercise, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. X ƒ(x) = ²x² = x 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Section 2.4
**Exercise 7**
For the following exercise, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
\[ f(x) = \frac{x}{x^2 - x} \]
---
#### Detailed Analysis:
To identify the points of discontinuity in the given function \( f(x) = \frac{x}{x^2 - x} \), follow these steps:
1. **Factor the denominator**:
\[
x^2 - x = x(x - 1)
\]
2. **Identify the points where the denominator is zero**:
\[
x(x - 1) = 0
\]
\[
x = 0 \quad \text{or} \quad x = 1
\]
At \( x = 0 \) and \( x = 1 \), the denominator becomes zero, making the function undefined at these points. These are potential points of discontinuity.
3. **Classify the discontinuities**:
- **At \( x = 0 \)**: The function has an infinite discontinuity because as \( x \) approaches 0, the function tends to infinity or negative infinity.
- **At \( x = 1 \)**: The function also has an infinite discontinuity because as \( x \) approaches 1, the function again tends to infinity or negative infinity.
By examining the function \( f(x) = \frac{x}{x(x - 1)} \), it's clear that the function is undefined at \( x = 0 \) and \( x = 1 \) due to division by zero, both leading to infinite discontinuities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf6bcca1-d77f-469d-8067-ac9f27c3130a%2Faf1f4c05-3891-4975-a5da-4a8a02ba9b62%2Fjd5z0n_processed.png&w=3840&q=75)
Transcribed Image Text:### Section 2.4
**Exercise 7**
For the following exercise, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
\[ f(x) = \frac{x}{x^2 - x} \]
---
#### Detailed Analysis:
To identify the points of discontinuity in the given function \( f(x) = \frac{x}{x^2 - x} \), follow these steps:
1. **Factor the denominator**:
\[
x^2 - x = x(x - 1)
\]
2. **Identify the points where the denominator is zero**:
\[
x(x - 1) = 0
\]
\[
x = 0 \quad \text{or} \quad x = 1
\]
At \( x = 0 \) and \( x = 1 \), the denominator becomes zero, making the function undefined at these points. These are potential points of discontinuity.
3. **Classify the discontinuities**:
- **At \( x = 0 \)**: The function has an infinite discontinuity because as \( x \) approaches 0, the function tends to infinity or negative infinity.
- **At \( x = 1 \)**: The function also has an infinite discontinuity because as \( x \) approaches 1, the function again tends to infinity or negative infinity.
By examining the function \( f(x) = \frac{x}{x(x - 1)} \), it's clear that the function is undefined at \( x = 0 \) and \( x = 1 \) due to division by zero, both leading to infinite discontinuities.
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
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
