Hi I am interested in a question regarding complex infinities. For example, consider the function
$$ q(x)=\frac{1}{x}\sqrt{i-1},\quad x\in \mathbb{R} $$ where $i=\sqrt{-1}$. Now let's take the limit of $q(x)$ as $x\to 0$, (from now on I will use $q(x)\equiv q$), $$ \lim_{x\to 0} q=\lim_{x\to 0}\left( \frac{1}{x}\sqrt{i-1}\right). $$ Is it appropriate to say $$ \lim_{x\to 0} q\to \infty? $$ What is the exact way to define the complex infinity? I am interested because I am taking a limit of $$ \lim_{x\to 0} \frac{qJ'_1(q)}{J_1(q)} $$ however I am unsure as to how to take the limit since I do not know how to deal with the complex infinities. If I understand how to deal with the complex infinity, I will be able to expand the Bessel functions as usual and then be able to take the limit. Thanks!
Also, in case you are interested $$ J'_1(q)=\frac{1}{2}(J_0(q)-J_2(q)) $$
Just use L'Hôpital's rule to get $1$ immediately. Seee https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule