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Chapter 1: Q. 6 (page 107)

Why do we have 0<|x-c|<未 instead of just |x-c|<未 in Definition 1.10?

Short Answer

Expert verified

As the absolute value of a number is always positive or zero, it is always equivalent to saying that|x-c|>0

Step by step solution

01

Step 1. Given Information  

The given statement is we have 0<|x-c|<未instead of just |x-c|<未in Definition 1.10

02

Step 2. Explanation

Definition 1.10 states that- The limit expression limx鈫�cf(x)=Lmeans that for all epsilon positive there exists delta positive such that if 0<|x-c|<未then|f(x)-L|<蔚

In this, the punctured interval is given by x鈭�(c-未,c)鈭�(c,c+未)which means that c-未<x<c+未

Now subtract c from each part,

c-未-c<x-c<c+未-c-未<x-c<未

This is the solution set for the inequality |x-c|<未

The fact thatx鈮�cmeans thatx-c鈮�0which is equivalent to saying thatx-c鈮�0and sincethe absolute value of a number is always positive or zero, it is always equivalent to saying that |x-c|>0

Thus, we must write0<|x-c|<未but not only|x-c|<未

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