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 day1by 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.
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