   Chapter 2.4, Problem 43E

Chapter
Section
Textbook Problem

Prove that lim x → 0 + ln x = − ∞

To determine

To prove: The limit of a function limx0+(lnx) is equal to by using the definition of a infinite limit.

Explanation

Definition of infinite limit:

“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 negative number N there is a positive number δ such that if 0<|xa|<δ then f(x)<N”.

Definition of Right-Hand limit:

“The limxa+f(x)=L if for every number ε>0 there is a number δ>0 such that if a<x<a+δ then |f(x)L|<ε”.

To guess: The number δ.

Let N be a given negative integer. Here a=0, f(x)=lnx.

By the definition of right hand limit and the infinite limit, it is enough to find a number δ>0 such that if 0<x<0+δ then lnx<N.

Consider lnx<N

lnx<Nx<eN

Therefore, there exists a number δ such that if 0<x0<δ then x<eN

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