limits of natural logarithm
The parent entry (http://planetmath.org/NaturalLogarithm) defines the natural logarithm as
and derives the
which implies easily by induction that
Basing on (1), we prove here the
Proof. By the above definition, is differentiable:
Accordingly, is also continuous and strictly increasing.
Let be an arbitrary positive number. We have . There exists a positive integer such that (see Archimedean property). By (2) we thus get , and since is strictly increasing, we see that
Hence the first limit assertion is true. Now . If , then and
(substitution (http://planetmath.org/SubstitutionForIntegration) ). From this we can infer the second limit assertion.
|Title||limits of natural logarithm|
|Date of creation||2014-12-12 10:15:50|
|Last modified on||2014-12-12 10:15:50|
|Last modified by||pahio (2872)|