EDIT: Problem solved; taking is (of course) just a mistake. However, I guess this thread still serves as a good exercise in proving multivariable limits by using polar coordinates. By the way, the original function in x and y is given by .

Consider the following solution to an exercise in which one is to give a proof using polar coordinates to prove the limit of a function at

Given

we want to show that the function has limitas approaches zero.

Assuming , we can establish the following,

Further, noting that, we have finally

Thus we may defineso that,

Now, here's my problem; I'm perfectly fine with the reasoning here up until the conclusion where is taken to be . To me it is fairly obvious that, instead, letting is a valid choice in establishing this proof, and that's what I've been teaching when instructing a few of the students who take a basic course in multivariable calculus at my university up until I was emailed this solution from a professor who teach the subject.

I don't see why taking could be correct, since with we'd be forced to take , but according to the definition we should be able to choose an arbitrary .

So, am I missing something here? Is there any reason why the first choice of should be valid or is it simply a typo?

Would be nice if someone could comment on this so that I can know if I'm really running around lying to the students around here or not.