Chapter 1: Q. 56 (page 98)

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.
$\underset{x \to 1^{+}}{l i m} \frac{1}{x^{2} - 1} = \infty, M = 10000, find largest δ > 0$

Short Answer

Expert verified

The required value of $δ \approx 0.00005$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \frac{1}{x^{2} - 1}$

Step 2. Explanation

From the given function, we have, $c = 1, M = 10000$

The limit expression can be written as a formal statement as below,

For allM positive, there exists a delta positive such that if $x \in (1, 1 + δ)$

Then $\frac{1}{x^{2} - 1} \in (10000, \infty)$

Now the largest value of delta is given by,

$\frac{1}{x^{2} - 1} = 10000 x^{2} - 1 = \frac{1}{10000} x^{2} = 0.0001 + 1 x = \sqrt{1.0001}$

Hence,

$δ = \sqrt{1.0001} - 1 \approx 1.0000499 - 1 \approx 0.0000499 \approx 0.00005$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Discrete Mathematics

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.limx→1+1x2-1=∞,M=10000,findlargestδ>0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Discrete Mathematics

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.