/*! 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. 12 Write each of the following ineq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q. 12 (page 107)

Write each of the following inequalities in interval notation:

0 < |x 鈭� c| < 未

Short Answer

Expert verified

Interval notation is(c鈭�未,c)鈭�(c,c+未)

Step by step solution

01

Step 1. Given Information:

Given inequality: 0 < |x 鈭� c| < 未

We want to write given inequalities in interval notation.

02

Step 2. Solution:

Formal Definition of Limit

The limit

limx鈫�cf(x)=Lmeansthatforall蔚>0,thereexists未>0suchthatifx鈭�(c鈭�未,c)鈭�(c,c+未),thenf(x)鈭�(L鈭�蔚,L+蔚).

Algebraic Definition of Limit

The limit

limx鈫�cf(x)=Lmeansthatforall蔚>0,thereexists未>0suchthatif0<|x鈭�c|<未,then|f(x)鈭�L|<蔚

The two definitions of limit are equivalent we get

(a)x鈭�(c鈭�未,c)鈭�(c,c+未)ifandonlyif0<|x鈭�c|<未;(b)f(x)鈭�(L鈭�蔚,L+蔚)ifandonlyif|f(x)鈭�L|<蔚.

By using (a) we have;

0<|x鈭�c|<未鈬�x鈭�x鈭�(c鈭�未,c)鈭�(c,c+未)

Therefore the interval notation is (c鈭�未,c)鈭�(c,c+未)

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

Find a formula for the cost Crof producing a gourmet soup can with radius rand height 5inches, and answer the following questions:

  1. What is the radius of a can that is 5inches tall and costs 30cents to produce?
  2. Your manager wants you to produce 5-inch-tall cans that cost between 20and 40cents. Write this requirement as an absolute value inequality.
  3. What range of radii would satisfy your manager? Write an absolute value inequality whose solution set lies inside this range of radii.

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

f(x)=(x-3),ifx<0-(x-3),ifx猢�0.

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�12x4=2;youmayassume未鈮�1.

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

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

limx鈫�03x2+1=1.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.