When some ratio has a limit in an “indeterminate
form”, in that , L’Hopital’s
rule (first discovered by Bernoulli) allows for calculating the
limit:

under certain conditions. And if at first you don’t succeed, try, try
again: when is also in
indeterminate form, you can differentiate as many times as needed.

As an aside, if we suppose that and can be approximated as Taylor series near , one can easily find that

when the leading terms of and are both zero, representing the
case where differentiations are needed. Here represents the
differentiation-with-respect-to- operator.

While I have occasionally run into cases where multiple applications
of L’Hopital’s rule are needed, the other day^{1}by
which I mean 7 years ago… I am a bit slow at updating this
blog I was struck by a situation which necessitated a
recursive application of L’Hopital’s rule. Consider the Lambert W
function which is defined implicitly by the equation

that is, is the inverse of the function . We will only be interested in the principal
real branch, which is defined for . Note that . Also, per Wikipedia, we have

Ok, now the calculation I ran into in my research was:

Let and so our goal is to calculate where the limit goes .

Note that . Starting by
applying the chain rule to the derivative of , we have

From what initially, as I was first calculating it, looked like it
would blow up endlessly, instead comes a surprisingly simple and clean
result!

Thanks Harry Altman for identifying a mistake in an earlier
draft.