b. Let Find and f(x) = if x > 0 if x = 0 -1 if x < 0. 1 0 lim f(x) 818 lim f(x). -O1E Justify your answer using the definition of a limit. (e.g. "Let e > 0 be given...")
b. Let Find and f(x) = if x > 0 if x = 0 -1 if x < 0. 1 0 lim f(x) 818 lim f(x). -O1E Justify your answer using the definition of a limit. (e.g. "Let e > 0 be given...")
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
![The text in the image is an exercise in calculus involving limits. It reads as follows:
---
**b. Let**
\[ f(x) = \begin{cases}
1 & \text{if } x > 0 \\
0 & \text{if } x = 0 \\
-1 & \text{if } x < 0
\end{cases} \]
Find
\[\lim_{x \to 0^-} f(x)\]
and
\[\lim_{x \to 0^+} f(x).\]
Justify your answer using the definition of a limit. (e.g.: "Let \(\epsilon > 0\) be given...")
---
### Explanation:
This exercise asks students to find the left-hand and right-hand limits of a piecewise function \( f(x) \) as \( x \) approaches 0 from both the negative (left) side and the positive (right) side. The function \( f(x) \) is defined in three parts, depending on the sign of \( x \).
The task is to determine:
1. \(\lim_{x \to 0^-} f(x)\): The limit of \( f(x) \) as \( x \) approaches 0 from the left (negative side).
2. \(\lim_{x \to 0^+} f(x)\): The limit of \( f(x) \) as \( x \) approaches 0 from the right (positive side).
Students are required to provide justification for their answer using the formal definition of a limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e25041d-7573-46df-b9d3-ec2dd7694c16%2F9b529e8a-0310-4648-b241-6b458da839a8%2Fvldj3kg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The text in the image is an exercise in calculus involving limits. It reads as follows:
---
**b. Let**
\[ f(x) = \begin{cases}
1 & \text{if } x > 0 \\
0 & \text{if } x = 0 \\
-1 & \text{if } x < 0
\end{cases} \]
Find
\[\lim_{x \to 0^-} f(x)\]
and
\[\lim_{x \to 0^+} f(x).\]
Justify your answer using the definition of a limit. (e.g.: "Let \(\epsilon > 0\) be given...")
---
### Explanation:
This exercise asks students to find the left-hand and right-hand limits of a piecewise function \( f(x) \) as \( x \) approaches 0 from both the negative (left) side and the positive (right) side. The function \( f(x) \) is defined in three parts, depending on the sign of \( x \).
The task is to determine:
1. \(\lim_{x \to 0^-} f(x)\): The limit of \( f(x) \) as \( x \) approaches 0 from the left (negative side).
2. \(\lim_{x \to 0^+} f(x)\): The limit of \( f(x) \) as \( x \) approaches 0 from the right (positive side).
Students are required to provide justification for their answer using the formal definition of a limit.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 19 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,
