   Chapter 2.4, Problem 38E

Chapter
Section
Textbook Problem

If H is the Heaviside function defined in Example 2.2.6, prove, using Definition 2, that lim t → 0 H ( t ) does not exist. [Hint: Use an indirect proof as follows. Suppose that the limit is L Take ε = 1 2 in the definition of a limit and try to arrive at a contradiction.]Definition 2 To determine

To prove: The limit of a function limt0H(t) does not exist.

Explanation

Definition of a limit:

Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then the limit of the f(x) as x approaches a is L, that is

limxaf(x)=L

If for every number ε>0 there is a number δ>0 such that

if 0<|xa|<δ then |f(x)L|<ε.

Proof:

Suppose that limt0H(t)=L.

By the definition of a limit,

Given ε=12 there exist δ>0 such that 0<|t|<δ implies

|H(t)L|<ε|H(t)L

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