/*! 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. 83 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. 83 (page 122)

Write a delta鈥揺psilon proof that shows that the function f(x)=xis continuous. You may find the following inequality useful: For any real numbers aand b, a-b鈮�a-b. (This exercise depends on Section 1.3.)

Short Answer

Expert verified

Hence we proved limx鈫�cx=c.

Step by step solution

01

Step 1. Given Information 

We are given the function f(x)=xis continuous and we need to write a delta鈥揺psilon proof.

02

Step 2. Delta–epsilon proof  

f(x)=xc=cL=c

For 蔚>0,未=蔚,

If 0<x-c<未we have,

x-c=x-c<未=蔚鈭�limx鈫�cx=c

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

For each limit statement in Exercises 41-44, use algebra to find 未or Nin terms of 蔚or M, according to the appropriate formal limit definition.

limx鈫�1-11-x=鈭�, find未in terms ofM.

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�3(x2-2x-3)=0;youmayassume未鈮�1

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�21x=12;youmayassume未鈮�1

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of fxas x鈫�cis the value fc.

(c) The limit of fxas x鈫�cmight exist even if the value of fcdoes not.

(d) The two-sided limit of fxas x鈫�cexists if and only if the left and right limits of fxexists as x鈫�c.

(e) If the graph of fhas a vertical asymptote at x=5, then limx鈫�5fx=鈭�.

(f) If limx鈫�5fx=鈭�, then the graph of fhas a vertical asymptote at x=5.

(g) If limx鈫�2fx=鈭�, then the graph of fhas a horizontal asymptote at x=2.

(h) Iflimx鈫�鈭�fx=2, then the graph offhas a horizontal asymptote aty=2.

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

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.