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Chapter 1: Q. 0 (page 106)

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Short Answer

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Limit limx→cf(x)=Lstate that the δ>0exists for all ε>0such that,

If 0<|x-c|<δ, then |f(x)-L|<ε.

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

(a)x∈(c-δ,c)∪(c,c+δ)ifandonlyif0<|x-c|<δ.(b)f(x)∈(L-ε,L+ε)ifandonlyif|f(x)-L|<ε.

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→cf(x)=Lstate that the δ>0exists for all role="math" localid="1648496107166" ε>0such that,

If 0<|x-c|<δ, then |f(x)-L|<ε.

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

(a)x∈(c-δ,c)∪(c,c+δ)ifandonlyif0<|x-c|<δ.(b)f(x)∈(L-ε,L+ε)ifandonlyif|f(x)-L|<ε.

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