Prove $\lim_{(x,y) \to (0,0)} \frac{\exp(xy)\cdot xy\cdot(x^2-y^2)}{x^2+y^2} =0$. I want to show that

$$\lim_{(x,y) \to (0,0)} \frac{\exp(xy)\cdot xy\cdot(x^2-y^2)}{x^2+y^2} =0.$$

Is it valid to do it like this:
$$\lim_{(x,y) \to (0,0)} \left|\frac{\exp(xy)\cdot xy\cdot(x^2-y^2)}{x^2+y^2}\right| \leq \lim_{(x,y) \to (0,0)} |\exp(xy)\cdot xy\cdot (x^2-y^2)|=0$$

We have that $e^{xy}\to 1$ and by polar coordinates 
$$\left|\frac{xy\cdot(x^2-y^2)}{x^2+y^2}\right|=r^2|(\cos \theta\sin \theta)(\cos^2\theta-\sin^2\theta)|\le r^2\to 0$$
