limits of natural logarithm

The parent entry ( defines the natural logarithmMathworldPlanetmathPlanetmath as

lnx=1x1tdt  (x>0) (1)

and derives the


which implies easily by inductionMathworldPlanetmath that

lnan=nlna. (2)

Basing on (1), we prove here the

Theorem.  The functionMathworldPlanetmathxlnx is strictly increasing and continuousMathworldPlanetmathPlanetmath on +.  It has the limits

limx+lnx=+andlimx0+lnx=-. (3)

Proof.  By the above definition, lnx is differentiableMathworldPlanetmathPlanetmath:

ddxlnx=1x> 0

Accordingly, lnx is also continuous and strictly increasing.

Let M be an arbitrary positive number.  We have  ln2=12dtt>0.  There exists a positive integer n such that  nln2>M (see Archimedean property).  By (2) we thus get  ln2n>M, and since lnx is strictly increasing, we see that


Hence the first limit assertion is true. Now  -M<0.  If  x>2n,  then  lnx>M  and

0<1x< 2-n,ln1x=11xdtt=x1duu=-lnx<-M

(substitution (  xt:=u).  From this we can infer the second limit assertion.

Title limits of natural logarithm
Canonical name LimitsOfNaturalLogarithm
Date of creation 2014-12-12 10:15:50
Last modified on 2014-12-12 10:15:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 33B10
Related topic ImproperLimits
Related topic GrowthOfExponentialFunction
Related topic FundamentalTheoremOfCalculusClassicalVersion
Related topic DifferentiableFunctionsAreContinuous