   Chapter 2.4, Problem 36E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Prove that lim x → 2 1 x = 1 2 .

To determine

To prove: The limit of a function limx2(1x) is equal to 12 by using the ε,δ definition of a 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, the limit of f(x) as x approaches a is L, 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”.

To guess: The number δ.

Let ε be a given positive integer. Here, a=2, L=12 and f(x)=1x.

By the definition of ε and δ, it is enough to find a number δ such that if 0<|x2|<δ then |1x12|<ε.

Consider |1x12|

|1x12|=|2x2x|=|2x||2x|=|x2||2x|

There exists a positive constant C, such that 1|2x|<C, then |x2|1|2x|<C|x2|<ε.

Thus, find a number δ such that 0<|x2|<δ then |x2|<εC.

So, choose δ=εC.

If x lies in any interval centered at 2, the value C is obtained as follows

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