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Chapter 1: Q. 0 (page 106)

Read the section and make your own summary of the material.

Short Answer

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Limit limx鈫�cf(x)=Lstate that the 未>0exists for all 蔚>0such that,

If 0<|x-c|<未, then |f(x)-L|<蔚.

The theorem of Equivalence of Geometric and Algebraic Definitions of the Limit state,

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

Step by step solution

01

Step 1. Given information.

Subchapter of delta epsilon proofs.

02

Step 2. Summary of chapter.

The algebraic definition of Limit is following.

Limit limx鈫�cf(x)=Lstate that the 未>0exists for all role="math" localid="1648496107166" 蔚>0such that,

If 0<|x-c|<未, then |f(x)-L|<蔚.

The theorem of Equivalence of Geometric and Algebraic Definitions of the Limit state the following statements.

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

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Most popular questions from this chapter

For each functionf graphed in Exercises23鈥�26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

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

limx鈫�12x+4=6.

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities:

Calculate each of the limits:

limx鈫�蟺2cotxcosx.

State what it means for a functionf to be continuous at a point x = c, in terms of the delta鈥揺psilon definition of limit.
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