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Chapter 1: Q. 1 (page 108)

Explain why it makes intuitive sense thatlimx鈫�c鈥�x=c for any real number c. Then use a delta鈥揺psilon argument to prove it.

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Step by step solution

01

Step 1. Given information

We have to explain why it makes intuitive sense that limx鈫�c鈥�x=cfor any real number c. Then use a delta鈥揺psilon argument to prove it.

02

Step 2. Explanation

Substitute c for x in the limit,

limx鈫�c鈥�x=c

For the limit statement limx鈫�c鈥�f鈥�(x)=L, the delta-epsilon statement is

For all 鈭�&驳迟;0, there exist 未>0such that whenever x鈭�(c鈭�未,c)鈭�(c,c+未)guaranteesf(x)鈭�(L鈭�鈭�,L+鈭�)

Fof every x satisfying 0<|x鈭�c|<未, every f(x) satisfies |f(x)鈭�L|<系

f(x)=xandL=c

Choose 未=鈭�

|f(x)鈭�L|=|x鈭�c|<未=鈭�

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