I have a question about something I’m wondering about. I’ve read somewhere that

L’Hopitals rule can also be applied to complex functions, when they are analytic.

So if have for instance:limz→0log(1+z)z?=limz→01(1+z)=1

Now i’m wondering if this is correct? Also if we take |z|<1, is it then correct?

Thanks,

**Answer**

L'Hopital's rule is a local statement: it concerns the behavior of functions near a particular point. The global issues (multivaluedness, branch cuts) are irrelevant. For example, if you consider limz→0, then it's automatic that only small values of z are in play. Saying "take |z|<1" is redundant.

Generally, you have a point a∈C and some neighborhood of a in which f,g are holomorphic. If f(a)=g(a)=0, then

limz→af(z)z−a=f′(a),limz→ag(z)z−a=g′(a)

hence

limz→af(z)g(z)=limz→af(z)/(z−a)g(z)/(z−a)=f′(a)g′(a)

Note that the above is a simple special case of the L'Hopital's rule, because we have (1). It's basically just the definition of derivative.

**Attribution***Source : Link , Question Author : user112167 , Answer Author : Post No Bulls*