Evaluate the following limit: $$\lim_{(x,y) \to (0,0)}\frac{-xy}{x^2 + y^2}$$
My try:
Using polar substitution - putting $x =r\cos(\theta)$, $y = r\sin(\theta)$,
$$\implies\lim_{r \to 0} \frac{-r^2\cos(\theta) \sin(\theta)}{r^2(\sin^2\theta+\cos^2\theta)} =\lim_{r \to 0} -\cos(\theta) \sin(\theta) = -\cos(\theta) \sin(\theta)$$
Does this make any sense?
Also, approaching from two different sides, $$\begin{align}\lim_{(x,y) \to (0,0)}\frac{-xy}{x^2 + y^2} & = \lim_{x\to 0}\frac{-x(0)}{x^2 + (0)^2} = 0\\\lim_{(x,y) \to (0,0)}\frac{-xy}{x^2 + y^2} & = \lim_{y\to 0}\frac{-(0)y}{(0)^2 + y^2} = 0 \end{align}$$
Both are equal... but the limit doesn't exist which I verified from Symbolab. What should I do to evaluate this?