$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(x)\sin(y)}{x^2+y^2}$ the limit is undefined.
I would like to know if the following method is sufficient to prove that the limit is undefined:
step1: $\lim_{(x,y)\rightarrow(x,0)}\frac{\sin(x)\sin(0)}{x^2+0^2}=0$
step2: $\lim_{(x,y)\rightarrow(0,y)}\frac{\sin(0)\sin(y)}{0^2+y^2}=0$
step3: but $\lim_{(x,y)\rightarrow(x,x)}\frac{\sin(x)\sin(x)}{x^2+x^2}=\frac{\sin(x)^2}{2x^2}=\frac{1}{2}(\frac{\sin x}{x})^2$
and as $x$ then $\rightarrow 0$ $\frac{\sin(x)^2}{2x^2}=\frac{1}{2}(\frac{\sin x}{x})^2 \rightarrow \frac{1}{2}$
hence the limit does not exist.
- Does this make sense?
- Does this method (the '3 steps') always work for functions of two variables?
- Would it be valid to use the same method to prove: $\lim_{(x,y)\rightarrow(0,0)}\frac{x^3-y^3}{x^2+y^2}=0$
Thank you!