Como puedo calcular este limite de 2 variables

$$\begin{align}&\lim_{(x,y) \to (0,0)} \frac{x^2-y^2}{\sqrt{x^2+y^2}}\\&\\&x=tu\\&y=tv\\&u^2+v^2=1\\&\lim_{t \to 0}=\frac{t^2u^2-t^2u^2}{\sqrt{t^2(u^2+v^2)}} =\lim_{t \to 0}  \frac{t^2(u^2-v^2)}{\sqrt{t^2}}\\&\lim_{t \to 0} t(u^2-v^2)=0\end{align}$$

Si el lim x, deberia tender a 0, veamos

$$\begin{align}&\bigg| \frac{x^2-y^2}{\sqrt{x^2+y^2}}\bigg|\leq \bigg| \frac{x^2-y^2}{\sqrt{x^2}} \bigg|\leq \bigg| \frac{x^2}{x} \bigg|=|x|\\&\lim_{(x,y) \to (0,0)}|x|=0\geq\bigg|\lim_{(x,y) \to (0,0)} \frac{x^2-y^2}{\sqrt{x^2+y^2}}\bigg|\\&\lim_{(x,y) \to (0,0)} \frac{x^2-y^2}{\sqrt{x^2+y^2}}=0\end{align}$$