   Chapter 2.4, Problem 16E

Chapter
Section
Textbook Problem

Prove the statement using the ε, δ definition of a limit and illustrate with a diagram like Figure 9.FIGURE9 lim x → 4 ( 2 x − 5 ) = 3

To determine

To prove: The statement limx4(2x5)=3.

Explanation

Definition used:

“Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then, the limit of 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”.

Let ε be a given positive integer. Here, a=4, L=3 and f(x)=2x5.

By the definition of ε and δ, it is enough to find a number δ such that, if 0<|x4|<δ, then |(2x5)3|<ε.

Consider |(2x5)3|.

|(2x5)3|=|2x53|=|2x8|=|2||x4|=2|x4|

Therefore, if 0<|x4|<δ, then 2|x4|<ε.

That is, if 0<|x4|<δ, then |x4|<ε2

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