I have a function in two variables $f(x, y)$ and need to calculate the limit $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} .$$ If I decide to change to polar coordinates, how can I determine where $r$ tends to?
I was thinking, since $r = \sqrt{x^{2} + y^{2}}$, on evaluating $$ \lim_{(x, y) \rightarrow (2, 3)}{r} = \lim_{(x, y) \rightarrow (2, 3)}{ \sqrt{x^{2} + y^{2}}} = \sqrt{13},$$ and then writing $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} = \lim_{r \rightarrow \sqrt{13}}{f(r \cos{\theta}, r \sin{\theta})} .$$
Is this correct?
It is not correct because $r=\sqrt{13}$ means all points that are $\sqrt{13}$ units away of the center. You can do a translation of the $R^2$ plane by the vector (2,3). Than f(x,y) would be f(x-2,y-3) and r->0.