Q. 71 Prove聽that聽limx鈫掆垶1x=0,聽wi... [FREE SOLUTION]

Chapter 1: Q. 71 (page 99)

$Prove that \lim_{x 鈫� 鈭�} \frac{1}{x} = 0, with these steps : (a) What is the 蔚鈥� N statement that must be shown to prove that \lim_{x 鈫� 鈭�} \frac{1}{x} = 0 ? (b) Argue that x 鈭� (N, 鈭�) if and only if x > N . (c) Argue that \frac{1}{x} 鈭� (0 鈭� 蔚, 0 + 蔚) if and only if 鈭� 蔚 < \frac{1}{x} < 蔚 . Then argue that for this limit, it suffices to consider 0 < \frac{1}{x} < 蔚 . (d) Given any particular 蔚, what value of N > 0 would guarantee that if x > N, then 0 < \frac{1}{x} < 蔚 ? Your answer will depend on 蔚 . (e) Put the previous four parts together to prove the limit statement .$

Short Answer

Expert verified

$(a) For all 蔚 > 0, there exists N > 0 such that if x 鈭� (N, 鈭�), then \frac{1}{x} 鈭� (0 鈭� 蔚, 0 + 蔚) . (b) x 鈭� (N, 鈭�) is equivalent to x > N . (c) \frac{1}{x} 鈭� (鈭� 蔚, 蔚) means that 鈭� 蔚 < \frac{1}{x} < 蔚 . For the limit we are considering, \frac{1}{x} approaches y = 0 from above, so, we actually wish to show that we can guarantee 0 < \frac{1}{x} < 蔚 . (d) For a given 蔚 > 0, choose N = \frac{1}{蔚} . (e) Given any 蔚 > 0, let N = \frac{1}{蔚} . If x 鈭� (N, 鈭�) then x > N > 0 by part (b) . Thus by our choice from part (d), 0 < \frac{1}{蔚} < x, and thus 0 < \frac{1}{x} < 蔚 . By part (c) this means 鈭� 蔚 < \frac{1}{x} < 蔚 and thus \frac{1}{x} 鈭� (0 鈭� 蔚, 0 + 蔚) .$

Step by step solution

Part (a) Step 1. The ε-N statement

$For all 蔚 > 0, there exists N > 0 such that if x 鈭� (N, 鈭�), then \frac{1}{x} 鈭� (0 鈭� 蔚, 0 + 蔚) .$

Part (b) Step 1. Argument regarding N

$Now, x 鈭� (N, 鈭�) is equivalent to x > N .$

Part (c) Step 1. Argument regarding ε

$Again, \frac{1}{x} 鈭� (鈭� 蔚, 蔚) means that 鈭� 蔚 < \frac{1}{x} < 蔚 . For the limit we are considering, \frac{1}{x} approaches y = 0 from above, so, we actually wish to show that we can guarantee 0 < \frac{1}{x} < 蔚 .$

Part (d) Step 1. Relationship between ε-N

$For a given 蔚 > 0, choose N = \frac{1}{蔚} .$

Part (e) Step 1. Proof of the limit statement

$Given any 蔚 > 0, let N = \frac{1}{蔚} . If x 鈭� (N, 鈭�) then x > N > 0 by part (b) . Thus by our choice from part (d), 0 < \frac{1}{蔚} < x, and thus 0 < \frac{1}{x} < 蔚 . By part (c) this means 鈭� 蔚 < \frac{1}{x} < 蔚 and thus \frac{1}{x} 鈭� (0 鈭� 蔚, 0 + 蔚) .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Part (a) Step 1. The ε-N statement

Part (b) Step 1. Argument regarding N

Part (c) Step 1. Argument regarding ε

Part (d) Step 1. Relationship between ε-N

Part (e) Step 1. Proof of the limit statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Decision Maths

Probability and Statistics

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.