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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Prove, using Definition 6, that lim x 3 1 ( x + 3 ) 4 =

Definition 6

Chapter 2.4, Problem 42E, Prove, using Definition 6, that limx31(x+3)4= Definition 6

To determine

To prove: The limit of a function limx3(1(x+3)4) is equal to by using the definition of a infinite limit.

Explanation

Definition used:

“Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then limxaf(x)= means that for every positive number M there is a positive number δ such that if 0<|xa|<δ then f(x)>M”.

To guess: The number δ.

Let ε be a given positive integer. Here, a=3 and f(x)=1(x+3)4.

By the definition of infinite limit it is enough to find a number δ such that if 0<|x(3)|<δ then 1(x+3)4>M

Consider 1(x+3)4>M

1(x+3)4>M1(x(3))4>M(x(3))4<1M|x(3)|4<1M

Take fourth root on both sides of the inequality,

|x(3)|44<1M4|x(3)|<1M4

Thus, find a number δ such that 0<|x(3)|<δ then |x(3)|<1M4

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