(3) It is a true fact that lim 142 5x – 7 The e-d proof for this limit given below contains a significant flaw in its reasoning. Find and explain the flaw, and then write a correct e-d proof for this limit. Incorrect proof: Let e > 0 be an arbitrary positive number, and let å = min{1, e}. Let r be a real number, and assume that 0 < |r – 2| < 8. Since 8 < 1, our assumption in particular means that 0 < |r – 2| < 1. From the latter inequality, we have: |r – 2| <1 = -1 1
(3) It is a true fact that lim 142 5x – 7 The e-d proof for this limit given below contains a significant flaw in its reasoning. Find and explain the flaw, and then write a correct e-d proof for this limit. Incorrect proof: Let e > 0 be an arbitrary positive number, and let å = min{1, e}. Let r be a real number, and assume that 0 < |r – 2| < 8. Since 8 < 1, our assumption in particular means that 0 < |r – 2| < 1. From the latter inequality, we have: |r – 2| <1 = -1 1
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
Transcribed Image Text:3
(3) It is a true fact that lim
= 1.
1+2 5x – 7
The e-d proof for this limit given below contains a significant flaw in its reasoning. Find and
explain the flaw, and then write a correct e-d proof for this limit.
Incorrect proof: Let e > 0 be an arbitrary positive number, and let å = min{1, ?e}. Let x be
a real number, and assume that 0 < |x – 2| < 8.
Since 8 < 1, our assumption in particular means that 0 < |r – 2| < 1. From the latter
inequality, we have:
|r – 2| <1 = -1< x – 2 < 1
> 1<r< 3
= 5< 5x < 15
> -2 < 5r – 7< 8
2 < [5x – 7| < 8
1
1
1
8
|5x – 72
1
To summarize, our assumption that |æ – 2| < 8 implies that
5x – 72
Since 8 <e, our assumption also implies that |r – 2| < e. Using these two facts together,
we have:
3
|3 — (5х — 7)
- 7
5x – 7
-
10 – 5x
5x – 7
= 5 |x – 2|
< 5 |r – 2|
(by the computation above)
2
1
< 5.
5 2
(since |r – 2| < e)
Expert Solution
Step 1
We have to find the incorrect step in the proof of the given limit which is
.
First observe the steps of the inequality . From this inequality, we
have to find the condition for .
Step by step
Solved in 3 steps
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,
