I don't quite understand one example in my notes it says.
My query is this: I don't understand what the significance of $\theta$ is. Why does it matter that $\theta \in (-\pi,\pi]$? I see the argument as this, clearly anyway in which we approach $0$ must each give the same limit otherwise the limit doesn't exist since the limit is independent of $w$ the limit only depends on $\theta$ hence for any two distinct values of $\theta$ of which there are infinitely many the limit is different $\implies$ the limit does not exist.
Is this correct?
Also what is $\theta$ changes as the complex number $w$ approaches zero what is to say we need to approach along a straight line with constant $\theta$?
Could anyone clear up my misconceptions?