2022 October 10

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.

by which I mean 7 years ago… I am a bit slow at updating this blog↩︎

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