• If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!


Detailed derivation for example of ln(x)

Page history last edited by Chris TC Wong 11 years, 11 months ago


there is a problem with the above example.

We may not know it is "legal" to evaluate the limit inside a function, unless we know for sure that the limit exists and the function is continuous.

Here we try not to prove that the function ln(x) is in fact continuous.

Instead, we make use of the inverse function of ln(x) is exp(x) and try to do the trick together with chain rule.

In the following demonstrations, let x>0.

Recall that Formula

so ,


we can show that


By Squeeze theroem,


Finally we use the inverse function of exp(x) to do the trick :


More troubles:

1. Why exp(x) is the inverse function of ln(x) or why ln(x) an inverse function of exp(x) ? 

2. What is chain rule?

3. Why I can switch the limit from h->0 to m->infinity?


Comments (0)

You don't have permission to comment on this page.