שלום
i was told here that there is no +infinity and -infinity in complex functions
only infinity
but i have a solved question from a test which is solved exactly like it you said was wrong
and got full credit
maybe i am missing something here
Really, it would have sufficed to note that since has a finite limit, and has an essential singularity at then has an essential singularity at , then what you showed (basically) is that the limit doesn't exist. I assume they gave you the credit since your argument vaguely appeals to what I said.
As an example of what you're doing wrong take . Clearly we can think of it as a real or complex function. Analyze the behaviour in both cases and remember why we can talk about poles.
Frankly I wouldn't know how to turn that expression into a Laurent series, but it's not necessary, that is why one develops propositions like: if with open, holomorphic and has a removable singularity at and has an essential singularity at then has an essential singularity at
has an essential singularity at infinity, which means has an essential singularity at (this is easy to check since the principal part of the Laurent expansion has infinitely many terms) and this gives that has an essential sing. wherever has a pole. Take and the conclusion follows.
Basically, because near a pole, a function behaves like for some integer m. Try to work out the details by yourself (maybe using the multiplication formula for will simplify the argument). For this specific case a simple change of variables (and maybe a little geometric argument) gives the conclusion.