I'm currently taking a calculus class and saw this question in previous finals.
Evaluate the following limit, as (x,y) goes to (0,0): $$\frac{x\sin y-y\sin x}{x^2+y^2} $$
The first thing I tried was using polar coordinates and the L'Hopitals rule to simplify the limit, since when we use polar coordinates, we can write $(r, \theta)$ goes to $(0, \theta)$ (At least that's what I thought). After turning the limit back to rectangular coordinates, the limit becomes this:
$$\frac{xy(\cos y-\cos x)}{(x^2+y^2)}$$ as $(x, y)$ goes to $(0, 0).$
Then I wrote $$x^2+y^2\ge y^2 \ge 0$$ $$2 \ge |\cos y-\cos x| \ge 0$$ $$|x||y| \ge 0$$
By using these three equations, I got: $$2\frac{|x||y|}{|x|^2+|y|^2} \ge \frac{|xy(\cos y-\cos x)|}{|x^2+y^2|} \ge 0$$
I aimed for left side of the inequality to be 0 when we take the limit, so that the Squeeze Theorem would work, but by Sertöz Theorem, left-hand side DNE. Until now, I have always used squeeze theorem and the build up to the solutions were similar to this. Are there any alternative solutions, or am I doing something wrong?