/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 89 Write a delta鈥揺psilon proof th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q. 89 (page 122)

Write a delta鈥揺psilon proof that proves that fis continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.

f(x)=x2

Short Answer

Expert verified

Ans: f(x)=x2is continuous in its domainx鈭�R.

Step by step solution

01

Step 1. Given information.

given,

f(x)=x2

02

Step 2. Domain

Since xis defined for allx鈭�R

03

Step 3. So, check for continuity.

Let cbe any real number.

fis continuous at x=c

assume that cis less than or equal to1

if, localid="1648043730196" limx鈫�c鈥�f(x)=f(c)

localid="1648043796713" LHS=limx鈫�c鈥�f(x)=limx鈫�c鈥�x2

by putting x=c

localid="1648044660051" =c2

localid="1648044676259" RHS=f(c)=c2

04

Step 4. Since, LHS =RHS

So, Function is continuous at x=c

Thus, we can write that

f(x)=x2is continuous for allx鈭�R

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

Most popular questions from this chapter

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.

Ifx>N, then1x2鈭�0<0.001

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�0(x3+1)=1

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

Each function in Exercises 9鈥�12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

limx鈫�-1-f(x)=2,limx鈫�-1+f(x)=2,f(-1)=1.

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47鈥�60.

limx鈫�1x2-1x-1=2
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.