Can I use l'H˘pital's rule for a complex function?
Lets say I want to investigate the behavior of function sin(z)/z around z=0, can I use it?
What do you mean by complex? That is not a complex function.
And You can use it in this case.
Since it is in undetermined form, ( 0 in the denominator and 0 in the numerator), you can take the derivative of the top and bottom, and then apply the limit again.
Well, Thank you very much but I know the result is 1.
My question is, can I use l'H˘pital's rule for this thing even due z is a number from the complex domain?
I mean when you try to find limit for this thing, there are number of paths you can take in order to do that, does l'H˘pital's rule works in that case?
Thanks a lot.
Sorry to raise a dead thread, but I have had this same question for some time and have never seen a proof of L'Hopital's Rule that does not require the mean value theorem for real differentiable functions, which as far as I have seen, does not extend to the complex numbers. Perhaps the mean value property of harmonic functions serves the same end, but I have still never come across such a proof (I must concede, I've never tried proving it myself-- it could turn out to be quite easy).
Evidence demonstrates that L'Hopital's Rule does apply to the limit of a complex valued analytic function (as I am not able to construct a counterexample), but I would love to see a proof if anyone knows where I can find one.
The proof is simple and relies on the factorization of holomorphic functions: Assume are holomorphic ( some disk around ). If then and where and , then the conclusion follows trivially. If on the other hand they both have a pole at then the it's a little trickier: Take and be the Laurent expansions around then the quotient of the functions are
and in both cases the limit is equal to: if , if and if .
Edit: As an afterthought notice that this proof relies on the fact that the functions are defined and not zero on a punctured neighbourhood of the limit point, so the general formulation, as in the case of a real variable, doesn't follow from this. So for example if you have one of the functions be such that doesn't extend in any way past any boundary point of a disk, then you can't use this to evaluate the limit at a boundary point.