Q. 88 Write a delta鈥揺psilon proof th... [FREE SOLUTION]

Chapter 1: Q. 88 (page 122)

Write a delta鈥揺psilon proof that proves that f is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.
$f (x) = x^{鈭� 1}$

Short Answer

Expert verified

Ans: $f (x) = x^{- 1}$ is continuous on its domain (continuous for all role="math" localid="1648040276972" $x 鈭� R - {0}$ )

Step by step solution

Step 1. Given information.

given, $f (x) = x^{鈭� 1}$

Step 2. Find Domain.

$f (x) = x^{- 1} = \frac{1}{x}$

At $x = 0$ ,

$f (0) = \frac{1}{0} = 鈭�$

Hence, is not defined at localid="1648042677677" role="math" $x = 0$

Step 3. So, check for continuity at all points except 0.

Let c be any real number except $0 .$

assume that $c$ is less than or equal to $1$

$f$ is continuous at $x = c$

if, $lim_{x 鈫� c} 鈥� f (x) = f (c)$

localid="1648049491586" $L H S = lim_{x 鈫� c} 鈥� f (x) = lim_{x 鈫� c} 鈥� \frac{1}{x} P u t t i n g x = c = \frac{1}{c}$

localid="1648049934543" $R H S = f (c) = \frac{1}{c}$

Step 4. LHS=RHS

The function is continuous at $x = c$ (Except $0$ )

Thus, we can write that

$f i s c o n t i n u o u s f o r a l l x 鈭� R - {0}$

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

Write a delta鈥揺psilon proof that proves that f is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.
$f (x) = x^{鈭� 1}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find Domain.

Step 3. So, check for continuity at all points except 0.

Step 4. LHS=RHS

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Mechanics Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Write a delta鈥揺psilon proof that proves that f is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.f(x)=x鈭�1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find Domain.

Step 3. So, check for continuity at all points except 0.

Step 4. LHS=RHS

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Mechanics Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Write a delta鈥揺psilon proof that proves that f is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.
$f (x) = x^{鈭� 1}$