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Chapter 1: Q. 9 (page 135)

Write the difference rule for limits in terms of delta鈥揺psilon statements.

Short Answer

Expert verified

The difference rule for limit is

limx鈫�c[f(x)-g(x)]=limx鈫�cf(x)-limx鈫�cg(x).

Step by step solution

01

Step 1. Given information.

We need to write the difference rule limx鈫�c[f(x)-g(x)]=limx鈫�cf(x)-limx鈫�cg(x)in terms of delta-epsilon statement.

02

Step 2. Proof.

For all 蔚>0, there exists 未>0such that if 0<|x-c|<未.

Then,

[f(x)-g(x)]-(L-M)<蔚f(x)-g(x)-L+M<蔚

where L=limx鈫�cf(x)and

M=limx鈫�cg(x).

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Most popular questions from this chapter

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 delta-epsilon proofs for each of the limit statements in Exercises 47鈥�60.

limx鈫�2x2-3x+2x-2=1

Write each of the inequalities in interval notation:

3x+1-2<0.1

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+1,ifx<13x-1,if1猢�x<2x+2,ifx猢�2.

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
See all solutions

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