Somehow (to me at least) related to lavinia's post is the interesting fact discovered by Euler
[tex] \mbox{for x very large} ~ \ln x \simeq \frac{1}{1} + \frac{1}{2} +\frac{1}{3} +...+\frac{1}{x} [/tex]
a result which can be recast rigorously in terms of limits.
