For a complex function which has an isolated singularity at

1. How do I find out if is a pole or not?

2. If it is a pole what is the order of the pole

Consider for e.g. . Is pole? If yes what is the order of the pole?

Is there a way to find answer to the above question without really writing down the Laurent Series about

A related question I would have is - do we have similar method to comment on essential and removable singularities?

Thanks